1. The transport PDE, du has a family of solutions u = f(z – vt) where f can be any function. (a) Show that the proposed solution obeys the wave oquation (b) Find f when u(r,0) = e=='/2; (c) Write down the solution u(z,t) when u(z,0) = e~x*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v =1
1. The transport PDE, du has a family of solutions u = f(z – vt) where f can be any function. (a) Show that the proposed solution obeys the wave oquation (b) Find f when u(r,0) = e=='/2; (c) Write down the solution u(z,t) when u(z,0) = e~x*/2; (d) Sketch the solution for t = 0, t = 1 and t = 5 for v =1
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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![1. The transport PDE,
du
du
has a family of solutions u = f(z – vt) where f can be any function.
(a) Show that the proposed solution obeys the wave oquation
(b) Find f when u(r,0) = e==*/2;
(c) Write down the solution u(z, t) when u(z,0) = e-*/2;
(d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1
2. The wave equation,
has a family of solutions u = f(x – vt) + g(z+ et) where ƒ can be any
function EDIT: provided v = 1.
(a) Show that the proposed solution obeys the wave equation
(b) Find f and g when u(r,0) = e-*/² and it has zero initial velocity,
dule = 0.
i.e.,
(c) Write down the solution u(z,t) when u(z,0) = e~x*/2;
(d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe37ebe49-731c-47f9-a49b-3a5f656faaa2%2Fc4bcd06c-25c2-4929-9596-b830f202508a%2Fdhv8y4u_processed.png&w=3840&q=75)
Transcribed Image Text:1. The transport PDE,
du
du
has a family of solutions u = f(z – vt) where f can be any function.
(a) Show that the proposed solution obeys the wave oquation
(b) Find f when u(r,0) = e==*/2;
(c) Write down the solution u(z, t) when u(z,0) = e-*/2;
(d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1
2. The wave equation,
has a family of solutions u = f(x – vt) + g(z+ et) where ƒ can be any
function EDIT: provided v = 1.
(a) Show that the proposed solution obeys the wave equation
(b) Find f and g when u(r,0) = e-*/² and it has zero initial velocity,
dule = 0.
i.e.,
(c) Write down the solution u(z,t) when u(z,0) = e~x*/2;
(d) Sketch the solution for t = 0, t = 1 and t = 5 for v = 1
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