<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, 1996

                                      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, 1996               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, 1996
--------------------------------              --------------------------------
Common Stock, $.33 1/3 par value                           4,167,900 shares








                                     -1-
<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,                
                       1996 and 1995                                    3
                                                                      
                 Balance Sheets --                                    
                       December 31, 1996 and September 30,            
                       1996                                             4
                                                                      
                 Statements of Cash Flows --                          
                       Three Months Ended December 31, 1996           
                       and 1995                                         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       
                                           ---------------------------
                                                            
                                              1996             1995   
                                           -----------      ----------
<S>                                        <C>              <C>
Operating revenues                         $10,063,000      $7,358,000
                                                            
Operating costs:                                            
    Operating expenses                       6,800,000       5,619,000
    General and administrative                 320,000         383,000
    Depreciation                             1,865,000       1,300,000
                                           -----------      ----------
                                                            
                                             8,985,000       7,302,000
                                           -----------      ----------
                                                            
Income from operations                       1,078,000          56,000
Other income (expense):                                     
         Interest income                        25,000          57,000
         Other                                   8,000            -
         Interest expense                     (118,000)           -
         Gain on disposal of assets             18,000           7,000
                                           -----------      ----------
                                                            
Income before income tax                     1,011,000         120,000
                                                            
Income tax benefit (expense):                               
         Current                              (187,000)       (135,000)
         Deferred                             (167,000)         92,000
                                           -----------      ----------
                                          
                                              (354,000)        (43,000)
                                           -----------      ---------- 
                                          
Net income                                 $   657,000      $   77,000
                                           -----------      ----------
                                          
                                          
Net income per common share                       $.16            $.02
                                                  ====            ====
                                          
Weighted average equivalent shares        
    outstanding                              4,185,201       4,204,797
                                           ===========      ==========
</TABLE>                                  
                                          
                                          
See accompanying note to the financial statements.





                                     -3-
<PAGE>   4
                           DAWSON GEOPHYSICAL COMPANY

                                 BALANCE SHEETS


<TABLE>                                             
<CAPTION>                                           
                                                     December 31, 1996         September 30,1996
                                                    -------------------        -----------------
                                                      (UNAUDITED)              
<S>                                                     <C>                    <C>
ASSETS                                                                         
Current assets:                                                                
   Cash and cash equivalents                            $  1,127,000                $ 1,493,000
   Marketable securities                                     991,000                    988,000
   Accounts receivable                                     8,724,000                  6,161,000
   Income taxes receivable                                     6,000                    193,000
   Prepaid expenses                                          138,000                    148,000
                                                         -----------                -----------
                                                                               
           Total current assets                           10,986,000                  8,983,000
                                                         -----------                -----------
                                                                               
                                                                               
Property, plant and equipment                             55,918,000                 56,368,000
   Less accumulated depreciation                         (24,005,000)               (23,442,000)
                                                         -----------                ----------- 
                                                                               
         Net property, plant and equipment                31,913,000                 32,926,000
                                                         -----------                -----------
                                                                               
                                                        $42,899,000                 $41,909,000
                                                        ===========                 ===========
                                                                               
LIABILITIES AND STOCKHOLDERS' EQUITY                                           
Current liabilities:                                                           
     Current maturities of long-term debt               $   857,000                 $   857,000
   Accounts payable                                        2,807,000                  2,079,000
   Accrued liabilities:                                                        
      Payroll costs and other taxes                          232,000                    560,000
      Other                                                   90,000                    144,000
                                                         -----------                -----------
                                                                               
           Total current liabilities                       3,986,000                  3,640,000
                                                         -----------                -----------
                                                                               
Long-term debt, less current maturities                    4,643,000                  4,857,000
                                                                               
Deferred income taxes                                       775,000                     608,000
                                                                               
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, 4,167,900        
       and 4,149,050 shares issued and              
       outstanding                                        1,389,000                   1,387,000
   Additional paid-in capital                            17,048,000                  17,016,000
   Retained earnings                                      15,058,000                 14,401,000
                                                         -----------                -----------
                                                                                     
           Total stockholders' equity                     33,495,000                 32,804,000
                                                         -----------                -----------
                                                                                     
                                                         $42,899,000                $41,909,000
                                                         ===========                ===========
                                                       
Contingencies (See Note 3)                          
</TABLE>                                            
                                                    
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    
                                                       ----------------------
                                                                    
                                                          1996         1995   
                                                       ----------   ----------
<S>                                                    <C>          <C>
Cash flows from operating activities:                               
     Net income                                        $  657,000   $   77,000
                                                                    
     Adjustments to reconcile net income to net                     
         cash provided by operating activities:                     
     Depreciation                                       1,865,000    1,300,000
     Gain on disposal of assets                           (18,000)      (7,000)
     Non-cash interest income                                -         (30,000)
     Non-cash compensation                                  8,000         -
     Deferred income tax (benefit) expense                167,000      (92,000)
     Change in current assets and liabilities:                      
         Increase in accounts receivable               (2,563,000)    (505,000)
         Decrease in income taxes receivable              187,000      126,000
         Decrease in prepaid expenses                      10,000      104,000
         Increase (decrease) in accounts payable          728,000     (256,000)
         Decrease in accrued liabilities                 (382,000)    (109,000)
         Increase in federal and state income taxes                 
           payable                                           -           9,000
                                                       ----------   ----------
                                                                    
Net cash provided by operating activities                 659,000      617,000
                                                       ----------   ----------
                                                                    
Cash flows from investing activities:                               
     Proceeds from disposal of assets                      34,000        7,000
     Capital expenditures                                (868,000)  (1,719,000)
     Proceeds from sale of marketable securities             -         745,000
                                                       ----------   ----------
                                                                    
Net cash used in investing activities                    (834,000)    (967,000)
                                                       ----------   ---------- 
                                                                    
Cash flows from financing activities:                               
     Principal payments on debt                          (214,000)        -
     Proceeds from exercise of stock options               23,000         -   
                                                       ----------   ----------
                                                                    
Net cash provided by financing activities                (191,000)        -   
                                                       ----------   ----------
                                                                    
Net decrease in cash and cash equivalents                (366,000)    (350,000)
                                                                    
Cash and cash equivalents at beginning                              
     of period                                          1,493,000    1,671,000
                                                       ----------   ----------
                                                                    
Cash and cash equivalents at end of period             $1,127,000   $1,321,000
                                                       ==========   ==========
</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
period presented.  The results of operations for the three months  ended
December 31, 1996, are not necessarily indicative of the results to be expected
for the fiscal year.


2.     NOTES PAYABLE

As of April 1, 1996, the Company has two notes payable that exist under a loan
agreement with a bank.  The loan agreement consists of  (1) a revolving line of
credit of $5,000,000 to mature April 15, 1997 with funding availability
determined by a borrowing base calculation; and (2) a term note of $6,000,000
to mature March 15, 2003.  Both notes are secured by eligible accounts
receivable and equipment purchased from loan proceeds.  The loan agreement
contains various restrictive covenants and compliance requirements.  Among
others, the agreement requires that no liens exist upon any of the collateral
nor any vehicle owned by the Company.  The notes bear interest at the bank's
prime rate (8.25% at December 31, 1996).  The term note requires monthly
principal and interest payments.

During fiscal 1996 the Company was advanced $6,000,000 on the term note for the
purchase of capital equipment.  For the fiscal years 1997 through 2002, the
annual maturity is $857,000, and for fiscal 2003, the annual maturity will be
the balance.  As of February 7, 1997, the Company has not utilized the
revolving line of credit.


3.     CONTINGENCIES

On July 1, 1995, an accident involving an automobile owned by the Company
claimed the lives of four employees.  The Company is a defendant in a lawsuit
by the families of two of the employees whose deaths resulted from the
accident.  The families filed suit against the Company under the gross
negligence provisions of the Texas Workers' Compensation Act.  Accordingly, the
Company believes its exposure is limited to claimed exemplary damages of $36
million.  The litigation is currently in the discovery stage.  The Company has
approximately $12 million of insurance coverage available to provide against an
unfavorable outcome in this matter.  Due to the uncertainties inherent in
litigation, no absolute assurance



                                      -6-
<PAGE>   7
can be given as to the ultimate outcome of this suit.  However, the Company
believes, based on knowledge of the facts to date and consultation with its
legal advisors, that liabilities, if any, from this suit should not have a
material adverse effect on the Company's financial position.

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 operations, as
the Company believes it is adequately insured.











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


OVERVIEW

The Company's performance for the quarter ended December 31, 1996, as compared
to the same quarter of the prior year, reflects the benefit of five operating
crews versus four and improved conditions with regard to weather and permit
problems.  The contrast of the periods illustrates that in reviewing the
Company's financial statements, it should be noted that fluctuations in the
Company's results of operations may occur due to weather and other factors.


RESULTS OF OPERATIONS

The Company's operating revenues for the first quarter of 1997 totaled
$10,063,000 versus $7,358,000 for the same period of fiscal 1996, an increase
of 36.8%.  Combined with continued demand for  acquisition of 3-D seismic data,
the Company's additional capacity from capital expansion during fiscal 1996 of
new equipment and technological upgrades to existing equipment increased
utilization to five acquisition crews during the first quarter of fiscal 1997
as compared to four crews that were negatively impacted by inclement weather
and permit problems during the first quarter of fiscal 1996. Minimal revenues
were generated through the acquisition and processing of 2-D seismic data.

Operating expenses for the quarter ended December 31, 1996 totaled $6,800,000,
an increase of $1,181,000, or 21%, over the same period of fiscal 1996.
Operating expenses increased primarily as a result of increased personnel and
other expenses associated with the equipment acquisitions and technological
upgrades made during  fiscal 1996.

General and administrative expenses for the quarter ended December 31, 1996
totaled $320,000, a decrease of $63,000 over the same period of fiscal 1996.
General and administrative expenses totaled 3.2% of operating revenues for the
quarter ended December 31, 1996 versus 5.2% of operating revenues for the same
period of the prior year.

Depreciation for the quarter ended December 31, 1996 totaled $1,865,000, an
increase of $565,000 from the same quarter of fiscal 1996.  Depreciation
increased as a result of the capital expansion discussed below in "Liquidity
and Capital Resources."








                                      -8-
<PAGE>   9
Total operating costs for the first quarter of fiscal 1997 totaled $8,985,000,
an increase of 23%, over the first quarter of fiscal 1996 due to the factors
described above.  Income from operations increased to $1,078,000, 10.7% of
revenues, from $56,000 in the comparable period of the prior year.  This
increase is the direct result of the Company's operating expenses being
relatively fixed as compared to revenue trends.  Because of the high proportion
of relatively fixed total operating costs (including personnel costs for active
crews and depreciation costs), income from operations reflects the significant
negative effects on revenues of the largely uncontrollable factors of weather
and permit problems in the prior year.


LIQUIDITY AND CAPITAL RESOURCES

   Cash Flows

     Net cash provided by operating activities increased to $659,000 for the
quarter ended December 31, 1996 from $617,000 as compared to the same period of
the prior year due to the increase in net income to $657,000 from $77,000 and
the increase in depreciation combined with fluctuations in working capital
components.  An increase in activity resulted in an increase to accounts
receivable for the quarter ended December 31, 1996 as compared to the same
period of the prior year.  The incurrence of current maturities of long-term
debt resulted in an increase in accounts payable for the first quarter of
fiscal 1997 as compared to the same quarter of the prior year.

     Net cash used in investing activities decreased to $834,000 for the first
quarter of fiscal 1997 from $967,000 in the same period of fiscal 1996.

     Net cash provided by financing activities decreased from the prior year
due to principal payments on debt.  As of April 1, 1996, the Company has two
notes payable that exist under a loan agreement with a bank.  The loan
agreement consists of (1)a revolving line of credit of $5,000,00 to mature
April 15, 1997 and (2)a term note of $6,000,000 to mature March 2003.  Both
notes are secured by eligible accounts receivable and equipment purchased from
loan proceeds.  The term note was fully advanced during fiscal 1996 and as of
February 7, 1997, no advances had been made on the revolving line of credit.

   Capital Expenditures

     Capital expenditures of $15,597,000 during fiscal year 1996 in addition to
capital expenditures during fiscal 1995 and 1994 have positioned the Company to
supply market demand with technologically advanced 3-D data acquisition
recording systems and leading edge data processing capabilities.
     Capital expenditures of $868,000 for the quarter ended December 31, 1996
represent additional capacity of the existing crews and fleet replacements.
Future capital expenditures will depend on the Company's evaluation of
technological advancements.







                                      -9-
<PAGE>   10
   Capital Resources

     The Company believes that its capital resources, including the
availability of bank borrowings, and cash flow from operations are adequate to
meet its current operational needs and finance future capital needs as
determined by market demand and technological developments.










                                      -10-
<PAGE>   11
                                   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
                                           Treasurer, Chief Financial Officer



DATE:           February 7, 1997        
       ---------------------------------





                                      -11-
<PAGE>   12
                              INDEX TO EXHIBITS


<TABLE>
<CAPTION>
EXHIBIT
NUMBER                          EXHIBIT
-------                         -------
<S>                     <C>
  27                    Financial Data Schedule

</TABLE>