(a) A wave is represented by a combination of two travelling waves as y(x, t) = 2 (sin [2π(3x + t)] + sin [2π(3x − t)]) . Specify the wavelength and frequency. (b) The function u(x, t) satisfies the wave equation, ² u əx² 8² u Ət² 0 (−L < x < L, t > 0), where c is the wave speed and L is a constant, together with the conditions u(−L, t) = u(L, t) = 0 2 u(x,0) = x²e du Ət (x,0) = 0 (t > 0), (−L < x < L), (−L < x < L). (i) Specify the largest value of c for which d'Alembert's solution can be used to find u(0, 2). (ii) Consider the case with L = 2 and c = 1/2. Using d'Alembert's solution, calculate u(0, 2), giving your answer to 3 significant figures. (c) An earthquake beneath the sea generates surface waves that are measured by a distant weather station. At 14:00 GMT on Saturday the waves at the weather station are measured to have period 20s, and at 19:00 GMT that day the period of the waves has decreased to 17s. Assume that the deep water wave theory applies and that groups of waves leave the earthquake site at the same time and travel with their respective group velocities. Take g = 9.81ms-2. Find the distance of the earthquake site from the station, and the time at which the earthquake occurred.
(a) A wave is represented by a combination of two travelling waves as y(x, t) = 2 (sin [2π(3x + t)] + sin [2π(3x − t)]) . Specify the wavelength and frequency. (b) The function u(x, t) satisfies the wave equation, ² u əx² 8² u Ət² 0 (−L < x < L, t > 0), where c is the wave speed and L is a constant, together with the conditions u(−L, t) = u(L, t) = 0 2 u(x,0) = x²e du Ət (x,0) = 0 (t > 0), (−L < x < L), (−L < x < L). (i) Specify the largest value of c for which d'Alembert's solution can be used to find u(0, 2). (ii) Consider the case with L = 2 and c = 1/2. Using d'Alembert's solution, calculate u(0, 2), giving your answer to 3 significant figures. (c) An earthquake beneath the sea generates surface waves that are measured by a distant weather station. At 14:00 GMT on Saturday the waves at the weather station are measured to have period 20s, and at 19:00 GMT that day the period of the waves has decreased to 17s. Assume that the deep water wave theory applies and that groups of waves leave the earthquake site at the same time and travel with their respective group velocities. Take g = 9.81ms-2. Find the distance of the earthquake site from the station, and the time at which the earthquake occurred.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) A wave is represented by a combination of two travelling waves as
y(x, t) = 2 (sin [2π(3x + t)] + sin [2π(3x − t)]) .
Specify the wavelength and frequency.
(b) The function u(x, t) satisfies the wave equation,
² u
əx²
8² u
Ət²
0 (−L < x < L, t > 0),
where c is the wave speed and L is a constant, together with the
conditions
u(−L, t) = u(L, t) = 0
2
u(x,0) = x²e
du
Ət
(x,0) = 0
(t > 0),
(−L < x < L),
(−L < x < L).
(i)
Specify the largest value of c for which d'Alembert's solution
can be used to find u(0, 2).
(ii)
Consider the case with L = 2 and c = 1/2. Using
d'Alembert's solution, calculate u(0, 2), giving your answer to
3 significant figures.
(c) An earthquake beneath the sea generates surface waves that are
measured by a distant weather station. At 14:00 GMT on
Saturday the waves at the weather station are measured to have
period 20s, and at 19:00 GMT that day the period of the waves
has decreased to 17s. Assume that the deep water wave theory
applies and that groups of waves leave the earthquake site at the
same time and travel with their respective group velocities. Take
g =
9.81ms-2.
Find the distance of the earthquake site from the station, and the
time at which the earthquake occurred.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c0b2a3d-410e-4110-bb5c-52cbbf4f21f4%2F45612071-5fdc-46f3-88fc-84ffc5506ca2%2F3yk7gv_processed.png&w=3840&q=75)
Transcribed Image Text:(a) A wave is represented by a combination of two travelling waves as
y(x, t) = 2 (sin [2π(3x + t)] + sin [2π(3x − t)]) .
Specify the wavelength and frequency.
(b) The function u(x, t) satisfies the wave equation,
² u
əx²
8² u
Ət²
0 (−L < x < L, t > 0),
where c is the wave speed and L is a constant, together with the
conditions
u(−L, t) = u(L, t) = 0
2
u(x,0) = x²e
du
Ət
(x,0) = 0
(t > 0),
(−L < x < L),
(−L < x < L).
(i)
Specify the largest value of c for which d'Alembert's solution
can be used to find u(0, 2).
(ii)
Consider the case with L = 2 and c = 1/2. Using
d'Alembert's solution, calculate u(0, 2), giving your answer to
3 significant figures.
(c) An earthquake beneath the sea generates surface waves that are
measured by a distant weather station. At 14:00 GMT on
Saturday the waves at the weather station are measured to have
period 20s, and at 19:00 GMT that day the period of the waves
has decreased to 17s. Assume that the deep water wave theory
applies and that groups of waves leave the earthquake site at the
same time and travel with their respective group velocities. Take
g =
9.81ms-2.
Find the distance of the earthquake site from the station, and the
time at which the earthquake occurred.
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