Question

Determine whether each of the following functions is a solution of wave equation: a) u(x, t) = sin (x − at), b) u(x, t) = sin (x − at) + ln (x + at)

Answer #1

Verify by substitution that U(x,t) = A eB(x-vt) is a
solution to the wave equation when A and B are constants.

Determine whether the equation is exact. If it is exact, find
the solution. If it is not, enter NS.
(y/x+7x)dx+(ln(x)−2)dy=0, x>0
Enclose arguments of functions in parentheses. For example,
sin(2x).

Verify that u(x, t) = v(x + ct) + w(x − ct) satisfies the wave
equation for any twice differentiable functions v and w.

1. a) Show that u(x, t) = (x + t) 3 is a solution of the wave
equation utt = uxx.
b) What initial condition does u satisfy?
c) Plot the solution surface.
d) Using (c), discuss the difference between the conditions u(x,
0) and u(0, t).

Find a solution u(x, t) of the following problem utt = 2uxx, 0 ≤
x ≤ 2 u(0, t) = u(2, t) = 0 u(x, 0) = 0, ut(x, 0) = sin πx − 2 sin
3πx.

Verify that the function in the following question is the
solution of wave equation
w=f(u), where f is a differentiable function of u, and u =
a(x+ct), where a is a constant

Solve the wave equation:
utt = c2uxx, 0<x<pi, t>0
u(0,t)=0, u(pi,t)=0, t>0
u(x,0) = sinx, ut(x,0) = sin2x, 0<x<pi

Suppose u(t,x) and v(t,x ) is C^2 functions defined on R^2 that
satisfy the first-order system of PDE Ut=Vx,
Vt=Ux,
A.) Show that both U and V are classical solutions to the wave
equations Utt= Uxx.
Which result from multivariable calculus do you need to justify
the conclusion.
B)Given a classical sol. u(t,x) to the wave equation, can you
construct a function v(t,x) such that u(t,x), v(t,x)
form of sol. to the first order system.

Determine whether cos ( k x ) e x p ( − i ω t ) is
an acceptable solution to the time-dependent Schrödinger wave
equation.

7. A wave on a string has a
displacement according to the equation:
y(x,t) = 25.0 cm sin ((36.0/m)x –
(8.00/sec)t)
Determine the amplitude, frequency,
period, velocity, and wavelength of the wave. Also determine the
transverse velocity at t = 0.16 sec.

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