Exhibit 4.1


 


Execution
Version


 


FIRST AMENDMENT TO RIGHTS AGREEMENT


 


This First
Amendment to Rights Agreement, dated as of October 25, 2021 (this “Amendment”), to the Rights Agreement,
dated as of April 8, 2021 (the “Company Rights Agreement”), between Dawson Geophysical Company, a Texas
corporation (the “Company”), and American Stock Transfer & Trust Company, LLC, a New York limited liability
company, as rights agent (the “Rights Agent”).


 


RECITALS


 


WHEREAS, Wilks Brothers, LLC,
a Texas limited liability company (“Parent”), WB Acquisitions Inc., a Delaware corporation and a subsidiary
of Parent (“Merger Sub”), and the Company contemplate entering into an Agreement and Plan of Merger (as the
same may be amended, supplemented or otherwise modified from time to time, the “Merger Agreement”), pursuant
to which it is proposed that, among other things, (a) on the terms, and subject to the conditions, set forth in the Merger Agreement,
Parent will cause Merger Sub to commence a tender offer on behalf of Parent (as it may be amended from time to time as permitted by this
Agreement, the “Offer”) to purchase any and all of the outstanding shares of common stock (“Common
Stock”) of the Company, par value $0.01 per share, at a price of $2.34 per share of Common Stock, in cash, without interest,
and (b) following consummation of the Offer, Merger Sub will merge with and into the Company, with the Company surviving as a subsidiary
of Parent, pursuant to the provisions of the Texas Business Organizations Code, on the terms, and subject to the conditions set forth,
in the Merger Agreement (the “Merger”).


 


WHEREAS, pursuant to Section 27
of the Company Rights Agreement, for so long as the Rights are then redeemable, the Company may, and the Rights Agent shall, if the Company
so directs, supplement or amend any provision of the Company Rights Agreement in any respect without the approval of any holders of Rights
(as defined in the Company Rights Agreement) or holders of Common Stock.


 


WHEREAS, the Company desires
to amend the Company Rights Agreement to render the Rights inapplicable to the Offer, the Merger Agreement, the Merger and the other transactions
specifically contemplated by the Merger Agreement.


 


WHEREAS, the Board of Directors
of the Company has approved the execution, delivery and performance by the Company of, and the consummation of the transactions, including
the Offer and the Merger, contemplated by, the Merger Agreement and has determined that it is in the best interests of the Company and
its shareholders to amend the Company Rights Agreement as set forth below and has approved this Amendment and authorized its appropriate
officers to execute and deliver the same to the Rights Agent.


 


Agreement


 


NOW, THEREFORE, for good and
valuable consideration, the receipt and adequacy of which are hereby acknowledged, the parties hereto agree as follows:


 


		1.	Capitalized Terms. All capitalized, undefined terms used in this Amendment shall have the meanings
assigned thereto in the Company Rights Agreement.



 






    	 	 	 



     


    

 


		2.	Amendments.



 


		a.	The definition of “Acquiring Person” set forth in Section 1 of the Company Rights Agreement
is hereby amended by adding the following sentences to the end of such definition:



 


“Notwithstanding anything in
this Agreement to the contrary, none of Wilks Brothers, LLC, a Texas limited liability company (“Parent”), WB
Acquisitions Inc., a Delaware corporation and a subsidiary of Parent (“Merger Sub”), or any of their respective
Affiliates or Associates shall be deemed to be an Acquiring Person, either individually or collectively, by virtue of or as a result of
(i) the approval, execution or delivery of the Agreement and Plan of Merger (as the same may be amended from time to time, the “Merger
Agreement”), dated as of October 25, 2021, among the Company, Parent and Merger Sub, or the approval, execution and/or
delivery of any amendment thereto, (ii) the commencement of the tender offer that Parent caused Merger Sub to commence on behalf
of Parent, on the terms and subject to the conditions set forth in the Merger Agreement (as it may be amended from time to time as permitted
by the Merger Agreement, the “Offer”), to purchase any and all of the outstanding shares of Common Stock at
a price of $2.34 per share, in cash, without interest, (iii) the consummation of the Offer, (iv) the consummation of the merger
of Merger Sub with and into the Company on the terms, and subject to the conditions, set forth in the Merger Agreement (such merger is
referred to in this Agreement as the “Merger”), (v) the consummation of any other transactions contemplated
by the Merger Agreement, or (vi) the public announcement of any of the Merger Agreement, the Offer, the Merger or any other transactions
contemplated by the Merger Agreement. However, if the Acceptance Time (as defined in the Merger Agreement) does not occur in accordance
with the Merger Agreement, then the immediately preceding sentence shall be of no further force and effect. The Company shall give the
Rights Agent prompt written notice upon each of the consummation of the Merger Agreement, the commencement of the Offer, the consummation
of the Offer and the occurrence of the Effective Time, and the Rights Agent may rely on such notice in carrying out its duties under this
Agreement and shall be deemed not to have any knowledge of the events described in this sentence unless and until it shall have received
such notice.”


 


		b.	The definition of “Distribution Date” set forth in Section 1 of the Company Rights Agreement
is hereby amended by adding the following sentence to the end of such definition:



 


“Notwithstanding anything in
this Agreement to the contrary, no Distribution Date shall occur or be deemed to have occurred by virtue of or as a result of (i) the
approval, execution or delivery of the Merger Agreement, or the approval, execution or delivery of any amendment thereto, (ii) the
commencement of the Offer, (iii) the consummation of the Offer, (iv) the consummation of the Merger, (v) the consummation
of any other transactions contemplated by the Merger Agreement, or (vi) the public announcement of any of the Merger Agreement, the
Offer, the Merger or any other transactions contemplated by the Merger Agreement.”


 


		c.	The definition of “Expiration Date” set forth in Section 1 of the Company Rights Agreement
is hereby amended and restated in its entirety as follows:



 


““Expiration Date”
shall mean the earliest of (i) the Final Expiration Date, (ii) the time at which the Offer is consummated, (iii) the time
at which the Rights are redeemed as provided in Section 23 hereof, (iv) the time at which the Rights expire pursuant to Section 13(d) hereof
and (v) the time at which all Rights then outstanding and exercisable are exchanged pursuant to Section 24 hereof.”


 



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		d.	The definition of “Permitted Offer” set forth in Section 1 of the Company Rights Agreement
is hereby amended by adding the following sentence to the end of such definition:



 


“Notwithstanding anything in
this Agreement to the contrary, the Offer shall be deemed to be a Permitted Offer.”


 


		e.	The definition of “Stock Acquisition Date” set forth in Section 1 of the Company Rights
Agreement is hereby amended by adding the following sentence to the end of such definition:



 


“Notwithstanding anything in
this Agreement to the contrary, no Stock Acquisition Date shall occur or be deemed to have occurred by virtue of or as a result of (i) the
approval, execution or delivery of the Merger Agreement, or the approval, execution or delivery of any amendment thereto, (ii) the
commencement of the Offer, (iii) the consummation of the Offer, (iv) the consummation of the Merger, (v) the consummation
of any other transactions contemplated by the Merger Agreement, or (vi) the public announcement of any of the Merger Agreement, the
Offer, the Merger or any other transactions contemplated by the Merger Agreement.”


 


		f.	Section 7 of the Company Rights Agreement is hereby amended by adding the following provisions at
the end of such section as new subsections (g) and (h):



 


“(g) Notwithstanding anything
in this Agreement to the contrary, the provisions of Sections 7(a), (b), (c), (d), (e) and (f) shall not apply by virtue of
or as a result of (i) the approval, execution or delivery of the Merger Agreement, or the approval, execution and/or delivery of
any amendment thereto, (ii) the commencement of the Offer, (iii) the consummation of the Offer, (iv) the consummation of
the Merger, (v) the consummation of any other transactions contemplated by the Merger Agreement, or (vi) the public announcement
of any of the Merger Agreement, the Offer, the Merger or any other transactions contemplated by the Merger Agreement.”


 


“(h) Notwithstanding anything
in this Agreement to the contrary, except for those provisions herein that expressly survive the termination of this Agreement, this Agreement
shall terminate upon the earlier of the Expiration Date and such time as all outstanding Rights have been exercised, redeemed or exchanged
hereunder.”


 


		g.	Section 9 of the Company Rights Agreement is hereby amended by adding the following provision at
the end of such section as a new subsection (f):



 


“(f) Notwithstanding anything
in this Agreement to the contrary, the provisions of Section 9 shall not apply by virtue of or as a result of the commencement or
consummation of the Offer.”


 


		h.	Section 11 of the Company Rights Agreement is hereby amended by adding the following provision at
the end of such section as a new subsection (q):



 


“(q) Notwithstanding anything
in this Agreement to the contrary, the provisions of Section 11 shall not apply, and no adjustments shall be made pursuant to this
Section 11, by virtue of or as a result of (i) the approval, execution or delivery of the Merger Agreement, or the approval,
execution and/or delivery of any amendment thereto, (ii) the commencement of the Offer, (iii) the consummation of the Offer,
(iv) the consummation of the Merger, (v) the consummation of any other transactions contemplated by the Merger Agreement, or
(vi) the public announcement of any of the Merger Agreement, the Offer, the Merger or any other transactions contemplated by the
Merger Agreement.”


 






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		i.	Section 13 of the Company Rights Agreement is hereby amended by adding the following provision at
the end of such section as a new section (e):



 


“(e) Notwithstanding anything
in this Agreement to the contrary, this Section 13 shall not be applicable to (i) the approval, execution or delivery of the
Merger Agreement, or the approval, execution and/or delivery of any amendment thereto, (ii) the commencement of the Offer, (iii) the
consummation of the Offer, (iv) the consummation of the Merger, (v) the consummation of any other transactions contemplated
by the Merger Agreement, or (vi) the public announcement of any of the Merger Agreement, the Offer, the Merger or any other transactions
contemplated by the Merger Agreement.”


 


		j.	Section 25 of the Company Rights Agreement is hereby amended by adding the following provision at
the end of such section as a new subsection (c):



 


“(c) Notwithstanding anything
in this Agreement to the contrary, the Company shall not be required to give any notice contemplated by this Section 25 by virtue
of or as a result of (i) the approval, execution or delivery of the Merger Agreement, or the approval, execution and/or delivery
of any amendment thereto, (ii) the commencement of the Offer, (iii) the consummation of the Offer, (iv) the consummation
of the Merger, (v) the consummation of any other transactions contemplated by the Merger Agreement, or (vi) the public announcement
of any of the Merger Agreement, the Offer, the Merger or any other transactions contemplated by the Merger Agreement.”


 


		k.	Section 30 of the Company Rights Agreement is hereby amended by adding the following sentence at
the end thereof:



 


“Further, nothing in this Agreement
shall be construed to give any holder of Rights or any other Person any legal or equitable rights, remedies or claims under this Agreement
by virtue of or as a result of (i) the approval, execution or delivery of the Merger Agreement, or the approval, execution and/or
delivery of any amendment thereto, (ii) the commencement of the Offer, (iii) the consummation of the Offer, (iv) the consummation
of the Merger, (v) the consummation of any other transactions contemplated by the Merger Agreement, or (vi) the public announcement
of any of the Merger Agreement, the Offer, the Merger or any other transactions contemplated by the Merger Agreement.”


 


		3.	Governing Law. Section 32 (Governing Law) of the Company Rights Agreement shall apply to this
Amendment mutatis mutandis.



 


		4.	Counterparts; Electronic Transmission. This Amendment may be executed and delivered by facsimile,
PDF or similar electronic transmission method in any number of counterparts and each of such counterparts shall for all purposes be deemed
to be an original, and all such counterparts shall together constitute but one and the same instrument.



 






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		5.	Authority. Each party represents that such party has full power and authority to enter into this
Amendment and that this Amendment constitutes a legal, valid and binding obligation of such party, enforceable against such party in accordance
with its terms.



 


		6.	Successors and Assigns. All of the covenants and provisions of this Amendment by or for the benefit
of the Company or the Rights Agent shall bind and inure to the benefit of their respective successors and assigns.



 


		7.	Benefits of this Amendment. Nothing in this Amendment shall be construed to give to any Person
other than the Company, the Parent, the Merger Sub, the Rights Agent and the registered holders of the Right Certificates (and, prior
to the Distribution Date, the Common Stock) any legal or equitable right, remedy or claim under this Amendment; but this Amendment shall
be for the sole and exclusive benefit of the Company, the Parent, the Merger Sub, the Rights Agent and the registered holders of the Right
Certificates (and, prior to the Distribution Date, the Common Stock).



 


		8.	Severability. If any term, provision, covenant or restriction of this Amendment is held by a court
of competent jurisdiction or other authority to be invalid, void or unenforceable, the remainder of the terms, provisions, covenants and
restrictions of this Amendment shall remain in full force and effect and shall in no way be affected, impaired or invalidated.



 


		9.	Effectiveness. This Amendment shall be effective as of the date first written above and shall be
deemed effective prior to, and shall be subject to, the execution and delivery of the Merger Agreement; provided, however, that this Amendment
shall be null and void, automatically terminate and be of no further force or effect if the Acceptance Time (as defined in the Merger
Agreement) does not occur in accordance with the Merger Agreement. In the event this Amendment is deemed null and void due to the Acceptance
Time not occurring, the Company shall provide the Rights Agent with notice of the Acceptance Time not occurring promptly thereafter and
the Rights Agent shall not be deemed to have any knowledge of the Acceptance Time not occurring unless and until it shall have received
such notice. Except as and to the extent expressly modified by this Amendment, the Company Rights Agreement and the exhibits thereto,
shall remain in full force and effect in all respects. In the event of a conflict or inconsistency between this Amendment and the Company
Rights Agreement and the exhibits thereto, the provisions of this Amendment shall govern.



 


		10.	Certification. The undersigned, an appropriate officer of the Company, hereby certifies to the
Rights Agent on behalf of the Company that this Amendment is in compliance with Section 27 of the Company Rights Agreement.



 


[Signature Page Follows]


 






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IN WITNESS WHEREOF, the parties
hereto have caused this Amendment to be duly executed as of the day and year first above written.


 


	 	DAWSON GEOPHYSICAL COMPANY
	 	 
	 	 	By:	/s/ Stephen C. Jumper
	 	 	Name:	Stephen C. Jumper
	 	 	Title:	President and Chief Executive Officer
	 	 
	 	AMERICAN STOCK TRANSFER& TRUST COMPANY, LLC
	 	 
	 	 	By:	/s/ Michael A. Nespoli
	 	 	Name:	Michael A. Nespoli
	 	 	Title:	Executive Director




 


Signature
Page to First Amendment to Rights Agreement