Consider the following wave equation, ... Which of the given options is/are solution(s) to the given wave equation?

Consider the following wave equation,

\[ \frac{\partial^2 f(x,t)}{\partial t^2} = 10000 \frac{\partial^2 f(x,t)}{\partial x^2} \]

Which of the given options is/are solutions to the above wave equation?

• (A) \[ f(x,t)=e^{-(x-100t)^2}+e^{-(x+100t)^2} \]
• (B) \[ f(x,t)=e^{-(x-100t)} +\frac12e^{-(x+1000t)} \]
• (C) \[ f(x,t)=e^{-(x-100t)} +\sin(x+100t) \]
• (D) \[ f(x,t)=e^{j100\pi(-100x+t)} + e^{j100\pi(100x+t)} \]

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Step 1: Identify the Wave Speed

The given wave equation is \[ \frac{\partial^2 f}{\partial t^2} = 10000 \frac{\partial^2 f}{\partial x^2}. \] Since \[ 10000=100^2, \] the wave speed is \[ c=100. \] Hence, the general solution of the one-dimensional wave equation is \[ f(x,t)=F(x-100t)+G(x+100t), \] where \(F\) and \(G\) are any twice differentiable functions. Therefore, every valid solution must be expressible as a function of

• \(x-100t\)
• \(x+100t\)

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Step 2: Verify Each Option

Option (A)

\[ f(x,t) = e^{-(x-100t)^2} + e^{-(x+100t)^2} \] Observe that \[ F(u)=e^{-u^2}, \qquad G(v)=e^{-v^2}. \] Therefore, \[ f(x,t)=F(x-100t)+G(x+100t), \] which exactly matches the general solution.

✔ Option (A) is a solution.

Option (B)

\[ f(x,t) = e^{-(x-100t)} + \frac12e^{-(x+1000t)} \] The second term contains \[ x+1000t \] instead of \[ x+100t. \] Let \[ g=e^{-(x+1000t)}. \] Then \[ g_{tt} = 1000^2g = 10^6g. \] Also, \[ g_{xx}=g. \] Hence \[ 10000g_{xx} = 10^4g. \] Since \[ 10^6g \neq 10^4g, \] the wave equation is not satisfied.

✘ Option (B) is NOT a solution.

Option (C)

\[ f(x,t) = e^{-(x-100t)} + \sin(x+100t) \] The first term depends on \[ x-100t. \] The second term depends on \[ x+100t. \] Thus, \[ f(x,t) = F(x-100t) + G(x+100t), \] which matches the general solution.

✔ Option (C) is a solution.

Option (D)

\[ f(x,t) = e^{j100\pi(-100x+t)} + e^{j100\pi(100x+t)} \] Consider the first exponential. Its argument is \[ t-100x, \] which is **not** of the form \[ x-100t \quad\text{or}\quad x+100t. \] Differentiate: \[ f_{tt}=k^2e^{ku}, \] where \[ u=t-100x, \qquad k=j100\pi. \] Also, \[ f_{xx} = 10000k^2e^{ku}. \] Hence, \[ 10000f_{xx} = 10^8k^2e^{ku} \neq f_{tt}. \] Therefore the wave equation is not satisfied.

✘ Option (D) is NOT a solution.

Final Answer

✔ (A) is correct.
✔ (C) is correct.

Correct Options: \[ \boxed{A \text{ and } C} \]

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