we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on-l
we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on-l
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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Question
[Second Order Equations] How do you solve 2? ty
![we will use the method of separation of variables and Fourier
series to derive a solution formula for the wave equation on -l < x <l
utt - c²uxx = 0,
with periodic boundary conditions
and initial conditions
u (-l, t) = u(l, t),
ux (−l, t) = ux (l, t),
(1)
u (x,0) = (x),
ut (x,0) = (x).
1. Suppose we have a "separated" solution u (x, t) = X(x)T(t). By following the method in
class, show that there must exist a real number number A so that
X" (x) + AX (x) = 0,
T" (t) + c²AT (t) = 0.
2. Show that in order for X to satisfy the correct boundary conditions, we must have λ = (¹7) ²
for n = 0,1,2,... and that we can take X to be any linear combination of cos (¹) or
sin (¹7x).
ηπχ
3. Solve the equation for T(t) (as we did in lecture) and then (just as in lecture) sum over all
possible separated solutions to obtain a guess for the general form of the solution u.
4. By evaluating the formula (and the time derivative of the formula) at t = 0, relate this
general formula to the initial data & and 4. Using the formulas for Fourier series on
-l < x < l, show that you can arrange the constants in your formula so that the function
u determined by the formula obtains the initial data correctly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F568d0f1d-4332-4acf-9fb6-227d9ac03ee6%2F7574da7d-f4c7-45ae-87fa-3af77afd421d%2F05um5ffq_processed.png&w=3840&q=75)
Transcribed Image Text:we will use the method of separation of variables and Fourier
series to derive a solution formula for the wave equation on -l < x <l
utt - c²uxx = 0,
with periodic boundary conditions
and initial conditions
u (-l, t) = u(l, t),
ux (−l, t) = ux (l, t),
(1)
u (x,0) = (x),
ut (x,0) = (x).
1. Suppose we have a "separated" solution u (x, t) = X(x)T(t). By following the method in
class, show that there must exist a real number number A so that
X" (x) + AX (x) = 0,
T" (t) + c²AT (t) = 0.
2. Show that in order for X to satisfy the correct boundary conditions, we must have λ = (¹7) ²
for n = 0,1,2,... and that we can take X to be any linear combination of cos (¹) or
sin (¹7x).
ηπχ
3. Solve the equation for T(t) (as we did in lecture) and then (just as in lecture) sum over all
possible separated solutions to obtain a guess for the general form of the solution u.
4. By evaluating the formula (and the time derivative of the formula) at t = 0, relate this
general formula to the initial data & and 4. Using the formulas for Fourier series on
-l < x < l, show that you can arrange the constants in your formula so that the function
u determined by the formula obtains the initial data correctly.
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