Q find Soluti on Of ion or me Vibrating equati Violin Shing azucxit) 6² Q?U (xit 1) 2. 2. which has bounday cunditi unj uc0it)=0 ; u(Lit) =0 (2) %3D and initiae condition: (3) ƏZUCXID-Asin satz *B Sinflex 2t
Q find Soluti on Of ion or me Vibrating equati Violin Shing azucxit) 6² Q?U (xit 1) 2. 2. which has bounday cunditi unj uc0it)=0 ; u(Lit) =0 (2) %3D and initiae condition: (3) ƏZUCXID-Asin satz *B Sinflex 2t
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Solving the Vibrating Violin String Equation**
We aim to find the solution to the equation governing the vibrating violin string:
\[
\frac{\partial^2 u(x, t)}{\partial t^2} = c^2 \frac{\partial^2 u(x, t)}{\partial x^2} \quad (1)
\]
**Boundary Conditions:**
\[
u(0, t) = 0, \quad u(L, t) = 0 \quad (2)
\]
**Initial Condition:**
\[
\frac{\partial u(x, 0)}{\partial t} = 0, \quad \frac{\partial^2 u(x, 0)}{\partial t^2} = A \sin\left(\frac{3\pi x}{L}\right) + B \sin\left(\frac{6\pi x}{L}\right) \quad (3), (4)
\]
We need to express the general solution using a Fourier series:
\[
u(x, t) = \sum_{n=1}^{\infty} \left[ a_n \cos(\omega_n t) + b_n \sin(\omega_n t) \right] \sin\left(\frac{n\pi x}{L}\right) \quad (5)
\]
**Where:**
\[
\omega_n = \frac{cn\pi}{L}, \quad \text{ensures that Eq. (5) is a solution}
\]
**Solving by the Wave Equation:**
To determine the coefficients \( a_n \) and \( b_n \), evaluate the first and second time derivatives of Eq. (5) at \( t = 0 \). These should be equal to the equations given in (3) and (4), respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fcc62d3-a3e3-4e53-b591-f5755305bd07%2F2526e6fa-e820-4a80-8482-adf5427f59c8%2Fkvssl6k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving the Vibrating Violin String Equation**
We aim to find the solution to the equation governing the vibrating violin string:
\[
\frac{\partial^2 u(x, t)}{\partial t^2} = c^2 \frac{\partial^2 u(x, t)}{\partial x^2} \quad (1)
\]
**Boundary Conditions:**
\[
u(0, t) = 0, \quad u(L, t) = 0 \quad (2)
\]
**Initial Condition:**
\[
\frac{\partial u(x, 0)}{\partial t} = 0, \quad \frac{\partial^2 u(x, 0)}{\partial t^2} = A \sin\left(\frac{3\pi x}{L}\right) + B \sin\left(\frac{6\pi x}{L}\right) \quad (3), (4)
\]
We need to express the general solution using a Fourier series:
\[
u(x, t) = \sum_{n=1}^{\infty} \left[ a_n \cos(\omega_n t) + b_n \sin(\omega_n t) \right] \sin\left(\frac{n\pi x}{L}\right) \quad (5)
\]
**Where:**
\[
\omega_n = \frac{cn\pi}{L}, \quad \text{ensures that Eq. (5) is a solution}
\]
**Solving by the Wave Equation:**
To determine the coefficients \( a_n \) and \( b_n \), evaluate the first and second time derivatives of Eq. (5) at \( t = 0 \). These should be equal to the equations given in (3) and (4), respectively.
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