<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, 1998
                                                ------------------  

                                       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, 1998         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)

    208 S. Marienfeld, Midland, Texas                          79701      
 ----------------------------------------               ------------------
 (Address of principal executive offices)                    (Zip Code)

        (Registrant's telephone number, including area code) 915/682-7356

                                      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.

<TABLE>
<CAPTION>
           CLASS                             Outstanding at March 31, 1998    
--------------------------------             -----------------------------
<S>                                          <C>
Common Stock, $.33 1/3 par value                   5,351,000 shares
</TABLE>


                                       -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 31, 1998 and 1997                                     3

                 Balance Sheets --
                       March 31, 1998 and September 30,
                       1997                                                              4

                 Statements of Cash Flows --
                       Six Months Ended March 31, 1998
                       and 1997                                                          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        
                                            -----------------------------     -----------------------------

                                                1998            1997              1998              1997    
                                            ------------     ------------     ------------     ------------
<S>                                         <C>              <C>              <C>              <C>         
Operating revenues                          $ 13,557,000     $ 11,721,000     $ 27,344,000     $ 21,784,000

Operating costs:
    Operating expenses                         9,808,000        7,913,000       18,901,000       14,713,000
    General and administrative                   483,000          346,000          965,000          666,000
    Depreciation                               2,244,000        1,906,000        4,355,000        3,771,000
                                            ------------     ------------     ------------     ------------

                                              12,535,000       10,165,000       24,221,000       19,150,000
                                            ------------     ------------     ------------     ------------

Income from operations                         1,022,000        1,556,000        3,123,000        2,634,000

Other income (expense):
      Interest income                            223,000           56,000          370,000           81,000
      Interest expense                                --         (112,000)        (125,000)        (230,000)
      Gain on disposal of assets                   3,000          175,000          148,000          193,000
      Other income                                 8,000            1,000           23,000            9,000
                                            ------------     ------------     ------------     ------------


Income before income tax                       1,256,000        1,676,000        3,539,000        2,687,000

Income tax expense:
      Current                                   (281,000)        (413,000)        (947,000)        (600,000)
      Deferred                                  (158,000)        (173,000)        (292,000)        (340,000)
                                            ------------     ------------     ------------     ------------
                                                (439,000)        (586,000)      (1,239,000)        (940,000)
                                            ------------     ------------     ------------     ------------

Net income                                  $    817,000     $  1,090,000     $  2,300,000     $  1,747,000
                                            ============     ============     ============     ============

Net income per common share                 $        .15     $        .26     $        .46     $        .42
                                            ============     ============     ============     ============

Net income per common share--
      assuming dilution                     $        .15     $        .26     $        .45     $        .42
                                            ============     ============     ============     ============

Weighted average equivalent shares
    outstanding                                5,351,000        4,171,122        5,028,187        4,166,524
                                            ============     ============     ============     ============

Weighted average equivalent shares
      outstanding--assuming dilution           5,376,636        4,188,012        5,057,918        4,189,456
                                            ============     ============     ============     ============
</TABLE>

See accompanying notes to the financial statements.





                                      -3-
<PAGE>   4


                           DAWSON GEOPHYSICAL COMPANY

                                 BALANCE SHEETS


<TABLE>
<CAPTION>
                                                     March 31, 1998  September 30, 1997
                                                     --------------  ------------------
                                                       (UNAUDITED)
<S>                                                  <C>             <C>  
ASSETS
Current assets:
   Cash and cash equivalents                          $  3,508,000     $  4,774,000
   Marketable securities                                13,025,000        3,968,000
   Accounts receivable                                  10,315,000        8,724,000
   Income taxes receivable                                 520,000               --
   Prepaid expenses                                        555,000          288,000
                                                      ------------     ------------

           Total current assets                         27,923,000       17,754,000
                                                      ------------     ------------


Property, plant and equipment                           68,977,000       63,267,000
   Less accumulated depreciation                       (31,579,000)     (27,460,000)
                                                      ------------     ------------

         Net property, plant and equipment              37,398,000       35,807,000
                                                      ------------     ------------

                                                      $ 65,321,000     $ 53,561,000
                                                      ============     ============

LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities:
   Current maturities of long-term debt               $         --     $  1,690,000
   Accounts payable                                      1,588,000        3,956,000
   Accrued liabilities:
      Payroll and other taxes                              587,000          566,000
      Other                                                244,000          494,000
                                                      ------------     ------------

           Total current liabilities                     2,419,000        6,706,000
                                                      ------------     ------------

Long-term debt, less current maturities                         --        7,893,000

Deferred income taxes                                    1,709,000        1,417,000

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,
       5,351,000 and 4,199,250 shares
       issued and outstanding                            1,784,000        1,400,000
   Additional paid-in capital                           38,183,000       17,174,000
   Retained earnings                                    21,226,000       18,971,000
                                                      ------------     ------------

           Total stockholders' equity                   61,193,000       37,545,000
                                                      ------------     ------------

                                                      $ 65,321,000     $ 53,561,000
                                                      ============     ============
</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        
                                                                ----------------------------
                                                                    1998             1997   
                                                                ------------     -----------
<S>                                                             <C>             <C>         
Cash flows from operating activities:
         Net income                                             $  2,300,000     $ 1,747,000

         Adjustments to reconcile net income to net cash
            provided by operating activities:
         Depreciation                                              4,355,000       3,771,000
         Gain on disposal of assets                                 (148,000)       (193,000)
         Non-cash interest income                                    (31,000)         (6,000)
         Deferred income taxes                                       292,000         340,000
         Other                                                        83,000           8,000
         Change in current assets and liabilities:
         Increase in accounts receivable                          (1,591,000)     (1,079,000)
         Increase in prepaid expenses                               (267,000)       (129,000)
         Decrease (increase) in income taxes receivable             (520,000)        193,000
         Decrease in accounts payable                             (2,368,000)       (882,000)
         Increase (decrease) in accrued liabilities                 (229,000)         81,000
         Increase in federal and state income
            taxes payable                                                 --         162,000
                                                                ------------     -----------

Net cash provided by operating activities                          1,876,000       4,013,000
                                                                ------------     -----------

Cash flows from investing activities:
         Proceeds from disposal of assets                            240,000         285,000
         Capital expenditures                                     (6,154,000)     (1,636,000)
         Proceeds from maturity of marketable
            securities                                             5,500,000              --
         Proceeds from sale of marketable securities                      --         742,000
         Investment in marketable securities                     (14,523,000)     (3,377,000)
                                                                ------------     -----------

Net cash used in investing activities                            (14,937,000)     (3,986,000)
                                                                ------------     -----------

Cash flows from financing activities:
         Principal payments on debt                               (9,583,000)       (428,000)
         Proceeds from exercise of stock options                       7,000         157,000
         Issuance of common stock                                 21,371,000              -- 
                                                                ------------     -----------

Net cash provided by (used in) financing activities               11,795,000        (271,000)
                                                                ------------     -----------

Net decrease in cash and cash equivalents                         (1,266,000)       (244,000)

Cash and cash equivalents at beginning of period                   4,774,000       1,493,000
                                                                ------------     -----------

Cash and cash equivalents at end of period                      $  3,508,000     $ 1,249,000
                                                                ============     ===========
</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, 1998, 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 matures 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 has not utilized 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 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


                                      -6-
<PAGE>   7

Notes to Financial Statements (continued)

claims asserted against it in such suits and it intends to continue to
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.  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 is 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.       PUBLIC OFFERING

         On November 21, 1997, the Company completed an offering of 1,150,000
shares.  The proceeds of the offering were approximately $21,371,000 after
deducting costs payable by the Company.

5.       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  
                                                      ------------------------    ------------------------
                                                            December 31                   March 31        
                                                      ------------------------    ------------------------
                                                         1998          1997          1998          1997   
                                                      ----------    ----------    ----------    ----------
     <S>                                              <C>           <C>           <C>           <C>
     Numerator:
       Net income and numerator for basic
         and diluted net income per
         common share-income available
         to common stockholders                       $2,300,000    $1,747,000    $  817,000    $1,090,000
                                                      ----------    ----------    ----------    ----------

     Denominator:
       Denominator for basic net income
         per common share-weighted
         average common shares                         5,028,187     4,166,524     5,351,000     4,171,122

       Effect of dilutive securities-
         employee stock options                           29,731        22,932        25,636        16,890
                                                      ----------    ----------    ----------    ----------
       Denominator for diluted net
         income per common share-
         adjusted weighted average
         common shares and assumed
         conversions                                   5,057,918     4,189,456     5,376,636     4,188,012
                                                      ----------    ----------    ----------    ----------

     Basic net income per common share                $      .46    $      .42    $      .15    $      .26
                                                      ==========    ==========    ==========    ==========

     Diluted net income per common
       share                                          $      .45    $      .42    $      .15    $      .26
                                                      ==========    ==========    ==========    ==========
</TABLE>





                                      -7-
<PAGE>   8
                    MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                 FINANCIAL CONDITION AND RESULTS OF OPERATIONS


OVERVIEW

The results of operations for the six months ended March 31, 1998 compare
favorably to the same period of the prior year despite operating interruptions
due to weather and permitting problems, which negatively impacted the quarter
ended March 31, 1998.  The Company is experiencing favorable weather during the
third quarter of fiscal 1998.  In November 1997, the Company completed a
secondary public offering of 1,150,000 common shares with net proceeds of
$21,371,000.  In reviewing the Company's financial statements it should be
noted that quarterly fluctuations in the Company's results of operations can
occur due to weather, land use permitting and other factors.

RESULTS OF OPERATIONS

The Company's operating revenues for the first six months of 1998 totaled
$27,344,000 versus $21,784,000 for the same period of fiscal 1997, an increase
of 25.5%.  For the three months ended March 31, 1998, operating revenues
totaled $13,557,000 versus $11,721,000 for the same period of fiscal 1997, an
increase of 15.7%.  The increase for the six-month period reflects the
Company's added production capacity in response to continued strong demand for
3-D seismic services.  In August 1997, the Company added a new, sixth data
acquisition crew.  To further complement capacity, the Company has continually
added channels to each of its crews.  The increase in the quarter ended March
31, 1998 versus the same period of fiscal 1997 indicates the effect of weather
and permitting problems.

Operating expenses for the six months ended March 31, 1998 totaled $18,901,000,
an increase of $4,188,000, or  28.5%, over the same period of fiscal 1997.  For
the quarter ended March 31, 1998, operating expenses increased $1,895,000, or
23.9%.  Operating expenses increased primarily as a result of increased
personnel and other expenses associated with equipment acquisitions and
technological upgrades.   The Company experienced two "one time" events during
the first quarter of fiscal 1998.  The Company received insurance proceeds of
approximately $380,000 resulting from livestock induced damages to equipment,
which was recognized as expense in fiscal 1997; and incurred approximately
$160,000 in equipment rental expense.

General and administrative expenses for the six months ended March 31, 1998
totaled $965,000, an increase of $299,000 from the same period of fiscal 1997.
For the quarter ended March 31, 1998, general and administrative expenses
totaled $483,000, an increase of $137,000 from the same quarter of fiscal 1997.
The increase primarily reflects additional personnel required to support
expanding operations.  General and administrative expenses totaled 3.5% of
operating revenues for the six months ended March 31, 1998 versus 3.1% of
operating revenues for the same period of the prior year.

Depreciation for the six months ended March 31, 1998 totaled $4,355,000, an
increase of $584,000 from the same period of fiscal 1997.  For the quarter
ended March 31, 1998, depreciation increased $338,000, or 17.7%.  Depreciation
increased as a result of the capital expansion discussed below in "Liquidity
and Capital Resources."

Total operating costs for the first six months of fiscal 1998 totaled
$24,221,000, an increase of 26.5%, from the same period of fiscal 1997 due to
the factors described above.  For the quarter ended March 31, 1998, operating
costs increased 23.3% from the same period of the prior year.  These increases
are consistent with the high proportion of relatively fixed total operating
costs (including personnel costs and depreciation) in conjunction with the
addition of the Company's sixth crew in August 1997.  Due to the unfavorable
circumstances described above, income from operations of $1,022,000 represents
7.5% of revenues for the three months ended March 31, 1998 as compared to a
ratio of 13.3% for the

                                      -8-

<PAGE>   9
same period of the prior year.  For the six-month periods of fiscal years 1998
and 1997, the ratios of income from operations to revenues of 11.4% and 12.0%
respectively, indicate the ability of results from operations to absorb some
degree of negative impacts.

LIQUIDITY AND CAPITAL RESOURCES

Cash Flows

Net cash provided by operating activities for the six months ended March 31,
1998 of $1,876,000 as compared to $4,013,000 for the same period of the prior
year is a decrease of $2,137,000.  While net income and depreciation resulted
in increases for fiscal 1998 from fiscal 1997, the net decrease is due to
fluctuations in working capital components.

Net cash used in investing activities increased to $14,937,000 from $3,986,000
resulting from investment of offering proceeds and increased capital
expenditures in the first six months of fiscal 1998 as compared to the same
period of fiscal 1997.

The cash flows provided by financing activities for the quarter ended December
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.  In April of 1998 the Company announced the purchase
of a 3,100- channel Input/Output System Two Remote Recording System to replace
the Halliburton MDS-18X recording system.  The approximate cost of the new
system is $5,000,000.  Capital expenditures to date, as well as an additional
$7,800,000 committed for the remainder of fiscal 1998, are for various
additions and replacements of cables and geophones, the continuous effort to
sustain a safe fleet of vehicles, the next generation of mainframe computers
for data processing, and leasehold improvements for the move of our corporate
office.  Depreciation has increased as a new crew has been placed into service
each year for the past several years.

Capital Resources

The Company believes that its capital resources including its holdings of
marketable securities, the availability of bank borrowings, 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.




                                       -9-
<PAGE>   10


                                    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
                                                 Vice President and Chief 
                                                 Financial Officer



DATE:    April 22, 1998              
     ----------------------



                                      -10-




<PAGE>   11

                               INDEX TO EXHIBITS

<TABLE>
<CAPTION>

EXHIBIT NO.                   DESCRIPTION
-----------                   -----------

<S>                      <C>                       
   27                    Financial Data Schedule
</TABLE>