Struggling to find solution to 1D wave equation in the following form:

AI Thread Summary
The discussion focuses on solving the 1D wave equation using Euler's identity and the relationships between complex exponentials and trigonometric functions. Participants emphasize the importance of correctly identifying and differentiating constants in the general solution, as the same symbols can represent different values in various contexts. It is suggested to express the general solution with distinct primed constants to avoid confusion and facilitate finding relationships between them. The conversation highlights that sine and cosine functions are valid solutions to the equation, allowing for avoidance of complex numbers. Overall, clarity in notation and understanding of complex coefficients is crucial for solving the equation effectively.
Ibidy
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Homework Statement
Seperate 1D wave equation into time dependent and indipendent form and show solution takes the following trig form.
Relevant Equations
1D wave equaiton
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You should know a relationship between ##e^{ix}## and trig functions of x.
 
haruspex said:
You should know a relationship between ##e^{ix}## and trig functions of x.
I tried using eulers identity but i just end up with a mess of complex and none complex trig functions rather than what they want.
 
You want a real value for T(t) = C_1e^{i\lambda ct} + C_2e^{-i\lambda ct}. So C_1 = A + iB and C_2 must be complex conjugates. Try expanding <br /> (A + iB)(\cos \lambda ct + i\sin \lambda ct) + (A - iB)(\cos \lambda ct - i\sin \lambda ct) and see what you get.
 
Im not quite sure how or what you did but its closer to anything i was able to find.
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Can i ask what you did?
 
In your solution in post #1, A, B, C and D can be complex, but you know that T and ##\phi## are real. That should allow you to establish some relationships between your constants.
 
Ibidy said:
I tried using eulers identity but i just end up with a mess of complex and none complex trig functions rather than what they want.
Perhaps your difficulty lies in writing the same symbols for constants that are different. Equation (15) is
##T(t)=A\sin(c\lambda t)+B\cos(c\lambda t).## You chose to write the general solution you derived as ##T(t)=Ae^{i\lambda ct}+Be^{-i\lambda ct}.## Why did you assign ##A## to the positive exponential and ##B## to the negative instead of the other way around? You could also have chosen linear combinations of ##A## and ##B## in front of each exponential and you would still have a solution. Constants ##A## and ##B## in equation (15) do not have the same meaning as your ##A## and ##B##. The same is also true for ##C## and ##D##.

If I were you, I would write my general solution as ##T(t)=A'e^{i\lambda ct}+B'e^{-i\lambda ct}## and then find a relation between the primed and unprimed constants. Start with equation (15) and observe that $$\cos \!x=\frac{e^{ix}+e^{-ix}}{2}~;~~\sin \!x=\frac{e^{ix}-e^{-ix}}{2i}=-i\frac{e^{ix}-e^{-ix}}{2}.$$Then gather like exponential terms. It's not that much of a mess.
 
You can avoid complex numbers altogether
if you recognize that sine and cosine are solutions
of the separated 2nd-order ordinary differential equations,
since they are [also] functions that are proportional to minus-their-second-derivative
(as verified by substitution).

Otherwise, as @kuruman has noted,
you will have to work with generally distinct complex coefficients
between the different forms of the general solution.
 
kuruman said:
Perhaps your difficulty lies in writing the same symbols for constants that are different.
I think it more likely that @Ibidy understands that the two sets of A, B, C, D are different, but has missed that in general they are complex. Without that, it is not possible to transmute the one set into the other.
 
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