Question 1. Solve the following IVPS for the 2D wave on the rectangle0, 1] x [0, 1] (where we assume that the solution u is always zero on theboundary):1.UttΔυ= –10 sin(37x) sin(7y)Ut(r, Y, 0) = sin(r 2) sin(ry)u(x, y, 0)2. (for this one, you can use a computer algebra software to compute theintegrals)Utt =Δυu(r, y, 0) = x(x – 1)°y(y – 1)*uz(r, y,0)-= sin(Tx) sin(TY)

Question 1. Solve the following IVPS for the 2D wave on the rectangle [0, 1] x [0, 1] (where we assume that the solution u is always zero on the boundary): 1. Utt :Au u(x, y,0) = – 10 sin(37x) sin(7y) Uz (x, y, 0) = sin(7x) sin(Ty) 2. (for this one, you can use a computer algebra software to compute the integrals) Utt = Δυ u(r, y, 0) = x(x – 1)°y(y – 1)³ u(r, y, 0) = sin(rr) sin(ry)

Question 1. Solve the following IVPS for the 2D wave on the rectangle [0, 1] x [0, 1] (where we assume that the solution u is always zero on the boundary): 1. Utt :Au u(x, y,0) = – 10 sin(37x) sin(7y) Uz (x, y, 0) = sin(7x) sin(Ty) 2. (for this one, you can use a computer algebra software to compute the integrals) Utt = Δυ u(r, y, 0) = x(x – 1)°y(y – 1)³ u(r, y, 0) = sin(rr) sin(ry)

Related questions

Q: Plane waves in a box with periodic boundary conditions Consider plane waves &(r) = a exp(ik. r) in a…

Q: Two in-phase cell phone towers at x = −50 m and x = 50 m on the x-axis are emitting 3.0 MHz radio…

A: 1.The diagram for the above question is shown below:

Q: If p=A*exp(i[kx-ωt]) is a plane wave in a medium with a density of ρa, determine the acoustic…

Q: -iwit, The wave function is given by (x, t) = c[3e-i@it&1(x) + 4e-iw2t¢2(x)] where 1(x) and 2(x) are…

A: So we will basically calculate the average of energy by doing normalization. Then the square of the…

Q: A 110HE sinusoidal The elechnc fied points in = 26.5 M. wave propagahing in The z-direc hon he…

Q: P13.28 Normalize the set of functions $„(0) = eino, 0 ≤ 0 ≤ 2TT. To do so, you need to multiply the…

Q: What are the wave vectors at which a standing wave is found for the diatomic chain? at the 1st BZ…

A: In a diatomic chain of atoms , there are two modes of vibration, acoustic and optical seperated by…

Q: Q2: The x component of vector bis -27.8 m and the y component is +48.5 m. What are the magnitude of…

A: Given, bx = - 27.8 m by = 48.5m We need to calculate, Magnitude and Angle from +x axis. =?

Q: Consider the normalized wave function, 1 1 (x) = c₁4₁(x) +42₂(x)+ ₂(x), 1 1 √8 √√2 3 where ₁(x),…

Q: Select the correct equation which relates the sine function to a cosine function Select one: O…

A: We are given some equations regarding phase relationship between sine and cosine functions. We need…

Q: - M₁M₂w¹ – 2C(M₁ + M₂)² + 2C²(1 — cos Ka) = 0. (22) We can solve this equation exactly for w², but…

Q: Find the Fourier series of a periodic wave f(x+ f(x) = 2c if -L < x< 0, f(x) = 3c if 0 < x < L. 2L)…

A: Given: The function is given by: fx=2c if -L≤x<0fx=3c if 0≤x<L Introduction: A…

Q: 3. The ground state and first excited state of the SHO for angular frequency w are Yo (x) = a…

Q: sin kx 2) Consider the wavefunction is given y(x) =2N For what value of N %3D will it be normalized?…

A: Since we only answer up to 1 question, we will answer the first question only. Please resubmit the…

Question

100%

Expert Solution

Step 1

Given

$u_{t t} = Δ u$

$u (x, y, 0) = - 10 \sin (3 πx) \sin (ππy)$

$u_{t} (x, y, 0) = \sin (πx) \sin (πy)$

Step 2

$u (0, y, +) = u (1, y, +) = u (x, 0, +) = u (x, 1, +) = 0$

$u (x, y, +) = X (x) Y (y) T (t) \frac{T^{''}}{T} = \frac{X^{''}}{X} + \frac{Y^{''}}{Y}$

X(0)=0 and X(1)=0

Y(0)=0 and Y(1)=0

$X_{n} (x) = \sin (n πx) Y_{m} (y) = \sin (m πy)$

$\frac{T^{''}}{T} = - {(n π)}^{2} - {(m π)}^{2}$

$T (+) = A_{n, m} \cos (\sqrt{n^{2} + m 2} π^{+}) + B_{n, m} \sin (\sqrt{n^{2} + m 2} π^{+})$

Step 3

$u (x, y, +) = \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} [A_{n, m} \cos (\sqrt{n^{2} + m 2} π^{+}) + B_{n, m} \sin (\sqrt{n^{2} + m 2} π^{+})]$ $= \sin (n x π) \sin (m y π)$

Substitute n=3 and m=1,

$u (x, y, 0) = \sin (3 x π) \sin (πy)$ $u_{+} (x, y, 0) = \sin (πx) \sin (πy)$

$A_{3, 1} = - 10 B_{1, 1} = \frac{1}{\sqrt{2} π}$

$u (x, y, +) = - 10 \cos (\sqrt{10} π^{+}) \sin (3 x π) \sin (y π) + \frac{1}{\sqrt{2} π} \sin (\sqrt{2} π^{+}) \sin (πx) \sin (πy)$

Step by step

Solved in 5 steps

Similar questions

WAVE EQUATIONS = I cos? 0 Avisible v = f1 = w/k w = 2nf = 2n/T, k = 21/A = 390 – 700 nm d sin Opright = m1 (т + 0.5)1 Ipoiarized d sin Odark = B = 10 log a sin Odark = m. I, = 1.00 x 10-12 W/m² 2t %3D тлn OR (т + 0.5)Л. Vstring = /Fr/µ H = m/e fbeat = If2 – fil n1 sin 01 = n2 sin 02 Omin = 1.22 1/D An = 1/n v±v. f v 7 vs y = A sin(kx – wt + 4) c = 3.00 x 108 m/s Vn = c/n, I « (Атpltd)2 fshifted %3| Area Vsound - 343 m/s Show work (present solutions). 1. A 100 Watt incandescent bulb is about 5.0% efficient at converting electrical energy into radiant energy (i.e. it produces 5.0 W of light energy). Assume the bulb projects light equally in all directions and produces light that is randomly Unpolarized light source 8.0 m polarized (in other words the source is isotropic and very small distance between filters (not to scale) unpolarized). A pair of 30 polarizing filters is used to Observer examine the light at a First filter has electric position 8.0 meters away field transmission axis…
Prove that the spherical wave given by: ?(?,?) = (A ei k (r-vt))/r is indeed a solution of the three-dimensional wave equation Hint: Write the wave equation in spherical coordinates (therefore, you should use the Laplacian written in terms of r, ?, ?, and insert the solution given above into the wave equation.
i need clear ans by hand and solve very very fast in 20 min and thank you | DYBALA 7.22 The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by = V= gλ 2п + 2πΥ βλ where g acceleration of gravity, A = wavelength, p = density. Y surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves).
Let Ap be the difference between the phase shift a Helium-Neon laser beam expe- riences on traversing a given length of vacuum and on traversing the same length of plasma. What is Ad when the laser beam passes through 10 cm of plasma having a density of n = 102? m?? How could this be used as a density diagnostic? The wavelength of the He-Ne laser radiation is 632.8 nm
D(x,0) 0.5 -0.5 -10 -5 5 10 15 20 х (m) D(0,t) 0.5 -0.5 -1 -0.5 0.5 1 1.5 2 2.5 3 t (s) Write down the motion of the wave of the form D(x, t) = A sin(kx – wt+ Po) that %3D matches the above snapshot at time t = 0 and history at location x=0. D(x,t) = m) sin (2 t x/( | m ) -2πt/( s) + (w) a (w) a
4
What is the difference between a continuous spectrum and a discrete spectrum? O A discrete spectrum is represented by distinct spectral components. A continuous spectrum represents a sound that continues indefinitely. O A discrete spectrum represents a sound that is turned on and off. A discrete spectrum is represented by an area under a curve.