Consider the wave equation = Uu – Uzz = 0, u(t,x). (1) Given the (Lorentz) transform from t, r to ť', x': (t, x) → (t', x') 1 t = y(t – Bx), r' = y(x – Bt), y = - V1 - 32 (2) where B: real constant, 0 < B < 1, [better to use: u(t, r) = u(t(t', x'), x(t', x')) = u(t', x') instead of w(t', x') in the notation of the textbook). a) Write the differential operators 8/at and 8/dx interms of ť' and a' using the Chain rule: 8/dt = (t'/at)(8/t')+ (dx' /8t)(8/dx'), 8/dx = (dť /dx)(8/dt') + (dx' /dx)(8/dx'). b) Write this wave equation interms of the new variables ť', ď'. [You should conclude that this wave equation is covariant/invariant (not change its form) under this transform.]
Consider the wave equation = Uu – Uzz = 0, u(t,x). (1) Given the (Lorentz) transform from t, r to ť', x': (t, x) → (t', x') 1 t = y(t – Bx), r' = y(x – Bt), y = - V1 - 32 (2) where B: real constant, 0 < B < 1, [better to use: u(t, r) = u(t(t', x'), x(t', x')) = u(t', x') instead of w(t', x') in the notation of the textbook). a) Write the differential operators 8/at and 8/dx interms of ť' and a' using the Chain rule: 8/dt = (t'/at)(8/t')+ (dx' /8t)(8/dx'), 8/dx = (dť /dx)(8/dt') + (dx' /dx)(8/dx'). b) Write this wave equation interms of the new variables ť', ď'. [You should conclude that this wave equation is covariant/invariant (not change its form) under this transform.]
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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this lesson is PDE partial
![Consider the wave equation
u = Utt
- Urz = 0,
u(t, x).
(1)
Given the (Lorentz) transform from t, x to ť', x': (t, x) → (ť', x')
1
t = y(t – Bx), a' = r(x – Bt),
(2)
- 32
where B: real constant, 0 < B < 1, [better to use: u(t, x) = u(t(t', x'), x(t', x')) = u(t', x')
instead of w(t', x') in the notation of the textbook].
a) Write the differential operators 8/ot and ở/dx interms of ť and r' using the Chain
rule: 8/dt = (at /t)() + (x/ət)(0/x'), 0/dx = (dt/dx)(0/0t') + (ar'/dx)(8/dx').
b) Write this wave equation interms of the new variables t', r'. [You should conclude
that this wave equation is covariant/invariant (not change its form) under this transform.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00b96c49-5d1e-4400-8dd4-26cadd130718%2Fd0c42360-10c0-4755-82b6-d50fb3b2d71b%2Fi5187i_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the wave equation
u = Utt
- Urz = 0,
u(t, x).
(1)
Given the (Lorentz) transform from t, x to ť', x': (t, x) → (ť', x')
1
t = y(t – Bx), a' = r(x – Bt),
(2)
- 32
where B: real constant, 0 < B < 1, [better to use: u(t, x) = u(t(t', x'), x(t', x')) = u(t', x')
instead of w(t', x') in the notation of the textbook].
a) Write the differential operators 8/ot and ở/dx interms of ť and r' using the Chain
rule: 8/dt = (at /t)() + (x/ət)(0/x'), 0/dx = (dt/dx)(0/0t') + (ar'/dx)(8/dx').
b) Write this wave equation interms of the new variables t', r'. [You should conclude
that this wave equation is covariant/invariant (not change its form) under this transform.]
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