d) Y(x, Solve the equations of part (c) for and, and show that t)=[f(x-at) + f(x + at)].
d) Y(x, Solve the equations of part (c) for and, and show that t)=[f(x-at) + f(x + at)].
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 97E
Related questions
Question
Please answer d
![Show that the wave equation Ytt = a²Yrr where a is a constant, can be
reduced to the form Yuv = 0 by the change of variables u = x + at and v=x-at
a
b) Use the results in part (a) and show that Y(x, t) can be written as
Y(x, t) = o(x+at) + (x-at),
where and are arbitrary functions.
Now, consider the wave equation in part (a) in an infinite one dimensional
medium subject to initial conditions
Y(x,0) = f(x),
Y₁(x,0) = 0,
-∞<x<∞, t> 0.
Using the form of the solution obtained in part (b), show that and must satisfy
Jo(x) + y(x) = f(x),
['(x)-'(x) = 0.
d) Solve the equations of part (c) for and, and show that
Y(x, t) = [f(xat) + f(x+at)].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F696ac12c-0bf2-4609-bddf-9b47926e3c62%2Fvtk8qk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Show that the wave equation Ytt = a²Yrr where a is a constant, can be
reduced to the form Yuv = 0 by the change of variables u = x + at and v=x-at
a
b) Use the results in part (a) and show that Y(x, t) can be written as
Y(x, t) = o(x+at) + (x-at),
where and are arbitrary functions.
Now, consider the wave equation in part (a) in an infinite one dimensional
medium subject to initial conditions
Y(x,0) = f(x),
Y₁(x,0) = 0,
-∞<x<∞, t> 0.
Using the form of the solution obtained in part (b), show that and must satisfy
Jo(x) + y(x) = f(x),
['(x)-'(x) = 0.
d) Solve the equations of part (c) for and, and show that
Y(x, t) = [f(xat) + f(x+at)].
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