<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 December 31, 1997

                                               -----------------
                                       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 December 31, 1997          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.

                   CLASS                        Outstanding at December 31, 1997
     --------------------------------           --------------------------------
     Common Stock, $.33 1/3 par value                   5,351,000 shares





<PAGE>   2
                           DAWSON GEOPHYSICAL COMPANY

                                     INDEX


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

                  Statements of Operations --
                        Three Months ended December 31,
                        1997 and 1996                                                       3

                  Balance Sheets --
                      December 31, 1997 and September 30,
                        1996                                                                4

                  Statements of Cash Flows --
                      Three Months Ended December 31, 1997
                        and 1996                                                            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
                                                         December 31
                                              --------------------------------
                                                  1997                1996
                                              ------------        ------------
<S>                                           <C>                 <C>         
Operating revenues                            $ 13,787,000        $ 10,063,000

Operating costs:
    Operating expenses                           9,093,000           6,800,000
    General and administrative                     482,000             320,000
    Depreciation                                 2,111,000           1,865,000
                                              ------------        ------------

                                                11,686,000           8,985,000
                                              ------------        ------------

Income from operations                           2,101,000           1,078,000
Other income (expense):
         Interest income                           147,000              25,000
         Other                                      15,000               8,000
         Interest expense                         (125,000)           (118,000)
         Gain on disposal of assets                145,000              18,000
                                              ------------        ------------

Income before income tax                         2,283,000           1,011,000

Income tax expense:
         Current                                  (666,000)           (187,000)
         Deferred                                 (134,000)           (167,000)
                                              ------------        ------------

                                                  (800,000)           (354,000)

Net income                                    $  1,483,000        $    657,000
                                              ============        ============


Net income per common share                   $        .31        $        .16
                                              ============        ============

Weighted average equivalent shares
    outstanding                                  4,745,268           4,185,201
                                              ============        ============
</TABLE>



See accompanying notes to the financial statements.






                                      -3-


<PAGE>   4
                           DAWSON GEOPHYSICAL COMPANY

                                 BALANCE SHEETS


<TABLE>
<CAPTION>
                                                    December 31, 1997  September 30,1997
                                                    -----------------  -----------------
                                                       (UNAUDITED)
<S>                                                 <C>                <C>
ASSETS
Current assets:
   Cash and cash equivalents                          $  3,151,000        $  4,774,000
   Marketable securities                                12,982,000           3,968,000
   Accounts receivable                                  12,610,000           8,724,000
   Prepaid expenses                                        286,000             288,000
                                                      ------------        ------------

           Total current assets                         29,029,000          17,754,000
                                                      ------------        ------------


Property, plant and equipment                           67,254,000          63,267,000
   Less accumulated depreciation                       (29,404,000)        (27,460,000)
                                                      ------------        ------------

         Net property, plant and equipment              37,850,000          35,807,000

                                                      $ 66,879,000        $ 53,561,000
                                                      ============        ============

LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities:
     Current maturities of long-term debt             $       -           $  1,690,000
   Accounts payable                                      3,820,000           3,956,000
   Accrued liabilities:
      Payroll costs and other taxes                        274,000             566,000
      Other                                                837,000             494,000
                                                      ------------        ------------

           Total current liabilities                     4,931,000           6,706,000
                                                      ------------        ------------

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

Deferred income taxes                                    1,551,000           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                           38,180,000          17,174,000
   Retained earnings                                    20,433,000          18,971,000
                                                      ------------        ------------

           Total stockholders' equity                   60,397,000          37,545,000

                                                      $ 66,879,000        $ 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>
                                                                                         Three Months Ended
                                                                                            December 31
                                                                                  --------------------------------
                                                                                      1997                1996
                                                                                  ------------        ------------
<S>                                                                               <C>                 <C>
Cash flows from operating activities:
      Net income                                                                  $  1,483,000        $    657,000

      Adjustments to reconcile net income to net cash provided by (used in)
         operating activities:
      Depreciation                                                                   2,111,000           1,865,000
      Gain on disposal of assets                                                      (145,000)            (18,000)
      Non-cash interest income                                                         (22,000)                -
      Deferred income taxes                                                            134,000             167,000
      Other                                                                             49,000               8,000
      Change in current assets and liabilities:
         Increase in accounts receivable                                            (3,886,000)         (2,563,000)
         Decrease in income taxes receivable                                               -               187,000
         Decrease in prepaid expenses                                                    2,000              10,000
         Increase (decrease) in accounts payable                                      (136,000)            728,000
         Decrease in accrued liabilities                                              (629,000)           (382,000)
         Increase in federal and state income taxes
           payable                                                                     680,000                 -
                                                                                  ------------        ------------

Net cash provided by (used in) operating activities                                   (359,000)            659,000
                                                                                  ------------        ------------

Cash flows from investing activities:
      Proceeds from disposal of assets                                                 209,000              34,000
      Capital expenditures                                                          (4,276,000)           (868,000)
      Proceeds from maturity of marketable securities                                1,000,000                 -
      Investment in marketable securities                                           (9,992,000)                -
                                                                                  ------------        ------------

Net cash used in investing activities                                              (13,059,000)           (834,000)
                                                                                  ------------        ------------

Cash flows from financing activities:
      Issuance of common stock                                                      21,371,000                 -
      Principal payments on debt                                                    (9,583,000)           (214,000)
      Proceeds from exercise of stock options                                            7,000              23,000
                                                                                  ------------        ------------

Net cash provided by (used in) financing activities                                 11,795,000            (191,000)
                                                                                  ------------        ------------

Net decrease in cash and cash equivalents                                           (1,623,000)           (366,000)

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

Cash and cash equivalents at end of period                                        $  3,151,000        $  1,127,000
                                                                                  ============        ============
</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 ended
December 31, 1997, 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.







                                      -7-


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


OVERVIEW

The results of operations for the quarter ended December 31, 1997 compare
favorably to the same quarter of the prior year and are consistent with the
quarter ended September 30, 1997. 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 quarter of 1998 totaled
$13,787,000 versus $10,063,000 for the same period of fiscal 1997, an increase
of 37.0%. This increase 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.

Operating expenses for the quarter ended December 31, 1997 totaled $9,093,000,
an increase of $2,293,000, or 33.7%, over the same period of fiscal 1997.
Operating expenses increased primarily as a result of the increased personnel
required to field the Company's added capacity. 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 quarter ended December 31, 1997
totaled $482,000, an increase of 50.6% over the same period 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 quarter ended December 31, 1997 versus 3.2% of operating
revenues for the same period of the prior year.

Depreciation for the quarter ended December 31, 1997 totaled $2,111,000, an
increase of $246,000 from the same quarter of fiscal 1997. Depreciation
increased as a result of the capital expansion discussed below in "Liquidity
and Capital Resources."

Total operating costs for the first quarter of fiscal 1998 totaled $11,686,000,
an increase of 30%, over the first quarter of fiscal 1997 due to the factors
described 




                                      -8-
<PAGE>   9

above. Income from operations increased to $2,101,000, 15.2% of revenues, from
$1,078,000, 10.7% of revenues, in the comparable period of the prior year. This
increase illustrates the result of the Company's operating expenses being
relatively fixed as compared to revenue trends.

LIQUIDITY AND CAPITAL RESOURCES

Cash Flows

Cash flows from operating activities resulted in a net use of $359,000 for the
three months ended December 31, 1997 as compared to net cash provided of
$659,000 for the same period of fiscal 1997. The increase in net income to
$1,483,000 from $657,000 resulted in a net use of cash from operating activity
due primarily to the increase in accounts receivable generated by increased
revenues.

Net cash used in investing activities increased to $13,059,000 from $834,000
resulting from investment of offering proceeds and increased capital
expenditures in the first quarter of fiscal 1998 as compared to the first
quarter 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 August of 1997 the Company placed a sixth crew into
service. The cost of the new crew equipped with a 2,000 channel I/O System Two
RSR was approximately $6,000,000. Depreciation has increased as a new crew has
been placed into service each year for the past several years. Expenditures
made during the first quarter of fiscal 1998 consist of additions and
replacements of cables and geophones and the continuous effort to sustain a
safe fleet of vehicles.

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:  February 9, 1998
       ----------------



                                      -10-



<PAGE>   11
                               Index to Exhibits


<TABLE>
<CAPTION>
Exhibit No.                        Description
-----------                        -----------
    <S>                       <C>
    27                        Financial Data Schedule
</TABLE>