<PAGE>   1
                                    FORM 10-Q

                       SECURITIES AND EXCHANGE COMMISSION
                              WASHINGTON D.C. 20549

                                   (Mark One)

     (X) QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES
                              EXCHANGE ACT OF 1934

                  For the quarterly period ended March 31, 1999
                                                 --------------
                                       OR

     ( ) TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES
                              EXCHANGE ACT OF 1934

                  For the transition period from____________ to

       For Quarter Ended March 31, 1999   Commission File number 2-71058
                         --------------                          -------
                           DAWSON GEOPHYSICAL COMPANY
              ---------------------------------------------------- 
             (Exact name of Registrant as specified in its Charter)

                   TEXAS                           75-0970548
                   -----                           ----------
    (State or other jurisdiction of      (IRS Employer Identification No.)
     incorporation or organization)

          508 West Wall, Suite 800, Midland, Texas      79701
          ----------------------------------------      ------
          (Address of principal executive offices)     (Zip Code)

        (Registrant's telephone number, including area code) 915/684-3000
                                                             ------------
                                      NONE
                                      ----
 (Former Name, Former Address & Former Fiscal Year if changed since last report)

     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days. Yes X . No   .
                                             ---    ---
  Indicate the number of shares outstanding of each of the issuer's classes of
common stock, as of the latest practicable date.

            CLASS                       Outstanding at March 31, 1999 
--------------------------------        ----------------------------- 
Common Stock, $.33 1/3 par value               5,406,794 shares

                                       -1-
<PAGE>   2
                           DAWSON GEOPHYSICAL COMPANY

                                      INDEX



<TABLE>
<CAPTION>
                                                                                       Page No.
                                                                                       --------
<S>               <C>                                                                  <C>
Part I.           Financial Information:

                  Statements of Operations --
                        Three Months and Six Months
                        ended March, 1999 and 1998                                          3

                  Balance Sheets --
                        March 31, 1999 and September 30,
                        1998                                                                4

                  Statements of Cash Flows --
                        Six Months Ended March 31,
                        1999 and 1998                                                       5

                  Notes to Financial Statements                                             6

                  Management's Discussion and Analysis of
                        Financial Condition and Results of
                        Operations                                                          8


Part II.          Other Information
</TABLE>








                                       -2-
<PAGE>   3
                          PART I. FINANCIAL INFORMATION

                           DAWSON GEOPHYSICAL COMPANY

                            STATEMENTS OF OPERATIONS
                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                               Three Months Ended             Six Months Ended    
                                                     March 31                      March 31       
                                            --------------------------      ----------------------
                                                                                                  
                                                1999           1998          1999         1998    
                                            ----------     -----------   -----------  ------------
                                                                                                  
<S>                                         <C>            <C>           <C>          <C>         
Operating revenues                          $6,053,000     $13,557,000   $14,071,000  $27,344,000 
                                            ----------     -----------   -----------  ----------- 
Operating costs:                                                                                  
         Operating expenses                  5,308,000       9,808,000    11,741,000   18,901,000 
         General and administrative            494,000         483,000     1,040,000      965,000 
         Depreciation                        2,698,000       2,244,000     5,409,000    4,355,000
                                            ----------      ----------   -----------  ----------- 
                                             8,500,000      12,535,000    18,190,000   24,221,000 
                                            ----------      ----------   -----------  ----------- 
Income (loss) from operations               (2,447,000)      1,022,000    (4,119,000)   3,123,000 
                                            
Other income (expense):

         Interest income                       212,000         223,000       398,000      370,000       
         Interest expense                         -               -             -        (125,000)      
         Gain (loss) on disposal of assets     (23,000)          3,000       (12,000)     148,000
         Other income                           36,000           8,000        42,000       23,000
                                            ----------      ----------   -----------   ----------

Income (loss) before income tax             (2,222,000)      1,256,000    (3,691,000)   3,539,000     
     Income tax benefit (expense):
         Current                               845,000        (281,000)    1,461,000     (947,000)
         Deferred                              (84,000)       (158,000)     (205,000)    (292,000)
                                            ----------      ----------    ----------   ----------
                                               761,000        (439,000)    1,256,000   (1,239,000)
                                            ----------      ----------   -----------   ----------

Net income (loss)                          $(1,461,000)     $  817,000    $(2,435,000) $ 2,300,000
                                           ===========      ==========    ===========  ===========
Net income (loss) per common share               $(.27)           $.15          $(.45)        $.46
                                           ===========      ===========   ===========   ==========


Net income (loss) per common share-
         assuming dilution                       $(.27)           $.15          $(.45)        $.45
                                           ===========      ==========     ==========   ==========
                                                                                          


Weighted average equivalent common
    shares outstanding                       5,406,416       5,351,000      5,387,397    5,028,187 
                                           ===========     ===========    ===========   ==========

Weighted average equivalent common
         shares outstanding-assuming
         dilution                            5,406,416       5,376,636      5,387,397    5,057,918
                                           ===========     ===========    ===========   ==========
</TABLE>

See accompanying notes to the financial statements.





                                       -3-
<PAGE>   4
                           DAWSON GEOPHYSICAL COMPANY

                                 BALANCE SHEETS

<TABLE>
<CAPTION>
                                                     March 31  1999              September 30, 1998
                                                     --------------              ------------------
                                                       (unaudited)
<S>                                                  <C>                        <C>  
ASSETS
Current assets:
   Cash and cash equivalents                         $11,966,000                        $ 5,745,000
   Short-term investments                              6,486,000                          6,515,000
   Accounts receivable                                 6,460,000                         11,821,000
   Income taxes receivable                             1,484,000                          1,050,000
   Prepaid expenses                                      392,000                            416,000
                                                     -----------                        -----------

           Total current assets                       26,788,000                         25,547,000
                                                     -----------                        -----------

Property, plant and equipment                         72,084,000                         73,584,000
   Less accumulated depreciation                     (31,617,000)                       (27,672,000)
                                                     -----------                        -----------
           Net property, plant and equipment          40,467,000                         45,912,000
                                                     -----------                        -----------


                                                     $67,255,000                        $71,459,000
                                                     ===========                        ===========

LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities:
   Accounts payable                                  $   577,000                        $ 1,766,000
   Accrued liabilities:
      Payroll costs and other taxes                      373,000                            635,000
      Other                                               31,000                            810,000
                                                     -----------                        -----------

           Total current liabilities                     981,000                          3,211,000
                                                     -----------                        -----------

Deferred income taxes                                  2,811,000                          2,606,000

Stockholders' equity:
   Preferred stock - par value $1.00 per share;
         5,000,000 shares authorized, none outstanding         -                                  -
   Common stock - par value $.33 1/3 per share;
       10,000,000 shares authorized, 5,406,794
         and 5,361,000 shares issued and outstanding
         in 1999 and 1998, respectively                1,802,000                          1,787,000
   Additional paid-in capital                         38,497,000                         38,256,000
   Retained earnings                                  23,164,000                         25,599,000
                                                     -----------                        -----------
           Total stockholders' equity                 63,463,000                         65,642,000
                                                     -----------                        -----------

                                                     $67,255,000                        $71,459,000
                                                     ===========                        ===========
</TABLE>

Contingencies (See Note 3)

See accompanying notes to the financial statements.

                                       -4-

<PAGE>   5
                           DAWSON GEOPHYSICAL COMPANY

                            STATEMENTS OF CASH FLOWS
                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                                                                 Six Months Ended
                                                                                     March 31            
                                                                         --------------------------------
                                                                             1999                1998    
                                                                         -----------         ------------
<S>                                                                      <C>                <C> 
Cash flows from operating activities:
         Net income (loss)                                                $ (2,435,000)     $  2,300,000

Adjustments to reconcile net income to net cash provided by operating
  activities:
         Depreciation                                                        5,409,000         4,355,000
         Loss (gain) on disposal of assets                                      12,000          (148,000)
         Non-cash interest income                                              (11,000)          (31,000)
         Non-cash compensation                                                 256,000              --
         Deferred income taxes                                                 205,000           292,000
         Other                                                                 160,000            83,000
         Change in current assets and liabilities:
           Decrease (increase) in accounts receivable                        5,361,000        (1,591,000)
           Decrease (increase) in prepaid expenses                              24,000          (267,000)     
           Increase in income taxes receivable                                (434,000)         (520,000)
           Decrease in accounts payable                                     (1,189,000)       (2,368,000)
           Decrease in accrued liabilities                                  (1,041,000)         (229,000)
                                                                          ------------       -----------

Net cash provided by operating activities                                    6,317,000         1,876,000
                                                                          ------------       -----------

Cash flows from investing activities:
  Proceeds from disposal of assets                                              16,000           240,000
  Capital expenditures                                                        (134,000)       (6,154,000)
  Proceeds from maturity of short term investments                           1,000,000         5,500,000
   Investment in short term investments                                       (978,000)      (14,523,000)
                                                                          ------------       -----------

Net cash used in investing activities                                          (96,000)      (14,937,000)
                                                                          ------------       -----------

Cash flows from financing activities:
  Principal payments on debt                                                      --          (9,583,000)
  Proceeds from exercise of stock options                                         --               7,000
  Issuance of common stock                                                        --          21,371,000
                                                                          ------------       -----------

Net cash provided by financing activities                                         --          11,795,000
                                                                          ------------       -----------

Net increase (decrease) in cash and cash
  equivalents                                                                6,221,000        (1,266,000)

Cash and cash equivalents at beginning
  of period                                                                  5,745,000         4,774,000
                                                                          ------------      ------------

Cash and cash equivalents at end of period                                $ 11,966,000      $  3,508,000
                                                                          ============      ============
</TABLE>


See accompanying notes to the financial statements.


                                       -5-
<PAGE>   6
                           DAWSON GEOPHYSICAL COMPANY

                          NOTES TO FINANCIAL STATEMENTS


1.       OPINION OF MANAGEMENT

         Although the information furnished is unaudited, in the opinion of
management of the Registrant, the accompanying financial statements reflect all
adjustments (consisting only of normal recurring accruals) necessary for a fair
presentation of the financial condition and results of operations for the
periods presented. The results of operations for the three months and the six
months ended March 31, 1999, are not necessarily indicative of the results to be
expected for the fiscal year.

2.       NOTES PAYABLE

         In April 1997, the Company entered into a loan agreement, as amended
(the "Loan Agreement"), with a bank. The Loan Agreement consists of (1) a
revolving line of credit of $6,000,000 which matured on April 15, 1999, (2) a
term note in the aggregate principal amount of $6,000,000 bearing interest at
the bank's prime rate and which matures on March 25, 2003 and (3) a term note in
the aggregate principal amount of $5,000,000 bearing interest at the prime rate
as published in The Wall Street Journal and which matures on April 15, 2003. The
notes are secured by eligible accounts receivable and equipment purchased from
loan proceeds.

         On November 25, 1997, the Company repaid all outstanding principal and
interest on the two Term Promissory Notes which have no reborrowing capacity.
The Company did not utilize the revolving line of credit.

3.       CONTINGENCIES

         The Company is a defendant in two lawsuits pending in the 112th and
83rd District Courts of Pecos County, Texas (respectively, Cause No. 8812,
Ernestine Bernal, et al. Vs. Javier Antonio Orona, et al.; and Cause No.
P5565-83-CV, Carla Jaquez, et al. Vs. Javier Antonio, et al.) relating to a July
1995 accident involving a van owned by the Company which was used to transport
employees to various job sites and a non-Company owned vehicle. The accident
resulted in the deaths of four Company employees who were passengers in such
van. The Company is one of several named defendants in such suits. Other named
defendants include the estate of the deceased driver of such van, who was an
employee of the Company, the driver of such non-Company owned vehicle, who was
then an employee of the Company, the owner of such vehicle, and Ford Motor
Company, the manufacturer of the Company van involved in such accident. In
general, the claims against the Company include allegations of negligence, gross
negligence and/or intentional tort as a result of, among other things, the
Company's alleged failure to provide safe transportation for its employees and
to properly select, train and supervise the deceased driver of such van. The
plaintiffs in such suits are seeking actual damages from the defendants of $15.5
million, additional unspecified actual damages, prejudgment and post-judgment
interest and costs of suit as well as exemplary and punitive damages in an
amount not to exceed four times the amount of actual damages. The Company
believes that it has meritorious defenses to the claims asserted against it in
such suits and it intends to


                                       -6-

<PAGE>   7


Notes to Financial Statements (continued)


vigorously defend itself against such claims. In addition, the Company believes
that it has approximately $11 million of liability insurance coverage to
provided against an unfavorable outcome. Such suits are currently in the
discovery stage and the Company currently has pending before the court a motion
for summary judgment in Cause No. 8812 requesting that the Company be dismissed
from such suit based upon various legal theories. Such motion has not yet been
heard by the court. Due to the uncertainties inherent in litigation, no
assurance can be given as to the ultimate outcome of such suits or the adequacy
or availability of the Company's liability insurance to cover the damages, if
any, which may be assessed against the Company in such suits. A judgment
awarding plaintiffs an amount significantly exceeding the Company's available
insurance coverage could have a material adverse effect on the Company's
financial condition, results of operations and liquidity.

         The Company is party to other legal actions arising in the ordinary
course of its business, none of which management believes will result in a
material adverse effect on the Company's financial position or results of
operation, as the Company believes it is adequately insured.


4.       NET INCOME PER COMMON SHARE

         The following table sets forth the computation of basic and diluted net
income per common share:

<TABLE>
<CAPTION>
                                                               Six Months Ended                     Three Months Ended
                                                                  March 31                                March 31          
                                                           --------------------------           ---------------------------
                                                               1999           1998                   1999         1998
                                                               ----           ----                   ----         ----

<S>                                                        <C>            <C>                   <C>             <C>
Numerator:
  Net income (loss) and numerator for basic
    and diluted net income per
    common share-income available
    to common stockholders                                 $(2,435,000)   $ 2,300,000            $(1,461,000)   $   817,000     
                                                           -----------    -----------            -----------    -----------  
                                                                                                                             
Denominator:                                                                                                                 
  Denominator for basic net income                                                                                           
    per common share-weighted                                                                                                
    average common shares                                    5,387,397      5,028,187              5,406,416      5,351,000  
  Effect of dilutive securities-                                                                                             
    employee stock options                                        --           29,731                   --           25,636  
                                                           -----------    -----------            -----------    -----------  
  Denominator for diluted net                                                                                                
    income per common share-                                                                                                 
    adjusted weighted average                                                                                                
    common shares and assumed                                                                                                
    conversions                                              5,387,397      5,057,918              5,406,416      5,376,636  
                                                           -----------    -----------            -----------    -----------  
                                                                                                                             
Net income (loss) per common share                         $      (.45)   $       .46            $      (.27)   $       .15  
                                                           ===========    ===========            ===========    ===========  
                                                                                                                             
Net income (loss) per common                                                                                                 
  share--assuming dilution                                 $      (.45)   $       .45            $      (.27)   $       .15  
                                                           ===========    ===========            ===========    ===========  
</TABLE>
                                                                            








                                       -7-

<PAGE>   8
MANAGEMENT'S DISCUSSION AND ANALYSIS OF
FINANCIAL CONDITION AND RESULTS OF OPERATIONS


The following discussion should be read in conjunction with the Company's
financial statements. In addition, in reviewing the Company's financial
statements it should be noted that the Company's revenues directly relate to oil
and gas exploration and production activity and fluctuations in the Company's
results of operations can occur due to weather, land use permitting and other
factors. Decreases in oil and gas activities have adversely affected the demand
for the Company's services and the Company's results of operations.

This report may contain certain forward-looking information regarding the
Company, including projections, estimates, forecasts, plans and objectives.
Although management believes that all such statements are based upon reasonable
assumptions, no assurance can be given that the actual results will not differ
materially from those contained in such forward-looking statements.


Overview

The revenues of the Company continue to be severely impacted due to low crude
oil prices. Demand for the Company's services is expected to be directly related
to crude oil prices; however, the Company is prepared for some delay in demand
for its services as crude oil prices recover. The Company has taken cost
reduction measures in response to decreased demand for its services.


Results of Operations

The Company's operating revenues for the first six months of 1999 totaled
$14,071,000 versus $27,344,000 for the same period of fiscal 1998, a decrease of
49%. For the three months ended March 31, 1999, operating revenues totaled
$6,053,000 versus $13,557,000 for the same period of fiscal 1998, a decrease of
55%. Demand for the Company's services has been negatively impacted by low crude
oil prices. During the quarter ended December 31, 1998, the Company reduced the
number of operating crews from six to three. The Company operated three crews
throughout the quarter ended March 31, 1999.

Operating expenses for the six months ended March 31, 1999 totaled $11,741,000,
a decrease of 38% from the same period of fiscal 1998. For the quarter ended
March 31, 1999, operating expenses decreased from $9,808,000 to $5,308,000, or
46%. Operating expenses decreased as a result of decreased demand for the
Company's services.


                                       -8-

<PAGE>   9

Additional cost reduction measures, such as employee layoffs and salary
reductions, were implemented in January 1999. The Company has reduced the number
of employees from 377 at September 30, 1998 to 191 at March 31, 1999. The
Company has retained key field personnel in anticipation of increased demand.

General and administrative expenses for the six months ended March 31, 1999
totaled $1,040,000, an increase of 8% from the same period of fiscal 1998. For
the quarter ended March 31, 1999, general and administrative expenses totaled
$494,000, an increase of $11,000, or 2%, over the same period of fiscal 1998.
The increase reflects personnel and other expenses added throughout fiscal 1998
that were required to support expanding operations. Cost reduction measures were
implemented in January 1999 to reduce general and administrative expenses. Rent
expense is also reflected in the six month period and the quarter ended March
31, 1999 as the office building owned and occupied since 1960 became outgrown,
and the functions performed in that building were moved to a more accommodating,
leased facility in July 1998.

Depreciation for the six months ended March 31, 1999 totaled $5,409,000, an
increase of $1,054,000 from the same period of fiscal 1998. For the quarter
ended March 31, 1999, depreciation increased $454,000. Depreciation continued to
increase as a result of capital expansion during fiscal 1998. See the discussed
below in "Liquidity and Capital Resources."

Total operating costs for the first six months of fiscal 1999 totaled
$18,190,000, a decrease of 25%, from the same period of fiscal 1998 due to the
factors described above. For the quarter ended March 31, 1999, operating costs
decreased 32%. The 49% decrease of revenues as compared to the 25% decrease of
total operating costs for the six months ended March 31, 1999 reflects the high
proportion of relatively fixed total operating costs (including personnel costs
of active crews and depreciation costs) inherent in the Company.


Liquidity and Capital Resources

Cash Flows

Net cash provided by operating activities of $6,317,000 for the six months ended
March 31, 1999 reflects decreases in accounts receivable and current liabilities
which occurred due to reduced revenues and cost cutting measures. The Company
historically has not had need to establish a reserve for bad debts. Accounts
receivable that are determined uncollectible will be recognized in the period
first determined to be uncollectible. The increase in income taxes receivable in
fiscal 1999 is a result of the Company's recognition of a tax benefit generated
by the net loss for the six month period ended March 31, 1999.


                                       -9-

<PAGE>   10

Net cash used in investing activities decreased to $96,000 from $14,937,000
resulting from significantly decreased capital expenditures in fiscal 1999 as
compared to fiscal 1998; and, in fiscal 1998, the Company invested proceeds from
the November 1997 public offering in U.S. Treasury instruments.

The cash flows provided by financing activities for the six month period ended
March 31, 1998 represent the net of the offering proceeds reduced by the
retirement of debt.

Capital Expenditures

The Company continually strives to supply market demand with technologically
advanced 3-D data acquisition recording systems and leading edge data processing
capabilities. Depreciation has increased as a new crew has been placed into
service each year for the past several years. Capital expenditures for fiscal
1999 are not anticipated to be material. The Company will maintain equipment in
and out of service in anticipation of increased future demand of the Company's
services.


Capital Resources

The Company believes that its capital resources including its short-term
investments and cash flow from operations are adequate to meet its current
operational needs and finance capital needs as determined by market demand and
technological developments.

Litigation

The Company is a defendant in two lawsuits relating to a July 1995 accident
involving a van owned by the Company in which four Company employees died. The
Company believes that it has meritorious defenses to the claims asserted against
it in such suits. Further, while the plaintiffs seek damages in excess of the
Company's liability insurance policies, the Company believes that its liability
insurance should provide adequate coverage of the damages, if any, which may be
assessed against the Company in such litigation. Due to the uncertainties
inherent in litigation, no assurance can be given as to the ultimate outcome of
such suits or the adequacy or availability of the Company's liability insurance
to cover any such damages. A judgment awarding plaintiffs an amount
significantly exceeding the Company's available insurance coverage could have a
material adverse effect on the Company's financial condition, results of
operations and liquidity.

Year 2000

The Company utilizes software and technologies throughout its operations that
may be vulnerable to the date change in the year 2000. Identification,
assessment, and in some


                                      -10-

<PAGE>   11

cases, replacement of equipment that may be affected by the year 2000 is
underway. Software controlled by the Company, including its propriety seismic
processing package, has been tested successfully. Replacements and upgrades have
not been accelerated by the year 2000 issue and do not represent costs in
addition to normal operating expenditures. The Company has begun communications
with its significant suppliers to determine if those parties have appropriate
plans to remedy year 2000 issues when their systems interface with the Company's
systems or may otherwise impact the operations of the Company. However, there
can be no guarantee that the systems of other companies, on which the Company's
systems rely, will be timely converted or that a failure to convert by another
company or a conversion that is incompatible with the Company's systems would
not have a material adverse effect on the Company. To date, the Company has
spent approximately $20,000 primarily in the assessment of and testing for year
2000 compliance. Assessment will continue throughout fiscal 1999, with an
additional estimated cost of $30,000. Although the Company is not aware of any
material operational issues, there can be no assurance that there will not be a
delay in, or increased costs associated with, the implementation of the
necessary systems and changes to address the year 2000. A potential source of
risk includes, but is not limited to, the inability of principal suppliers to be
year 2000 compliant, which could result in an interruption of the Company's
services. The Company currently does not have a formal contingency plan. If
unforeseen problems are encountered that relate to the year 2000, possible
solutions will be evaluated and the most efficient will be enacted.
























                                      -11-
<PAGE>   12
                                    SIGNATURE




                              Pursuant to the requirements of the Securities
                              Exchange Act of 1934, the Registrant has duly
                              caused this report to be signed on its behalf by
                              the undersigned, thereunto duly authorized.


                                        DAWSON GEOPHYSICAL COMPANY  
                                        ---------------------------------------
                                        (REGISTRANT)



                                        By:  /s/ L. Decker Dawson          
                                             ----------------------------------
                                        L. Decker Dawson
                                        President



                                        /s/ Christina W. Hagan             
                                        ---------------------------------------
                                        Christina W. Hagan
                                        Chief Financial Officer



DATE:       April 23, 1999    
            --------------



                                      -12-
<PAGE>   13
                               INDEX TO EXHIBITS


                    Exhibit No.                  Description
                    -----------                  -----------
                        27                   Financial Data Schedule