Defining Wave Properties

Question 1

What is a wave?

Answer 1

An oscillation in time that propagates through space, carrying energy and information.

Question 2

What is the simplest equation of a wave?

Answer 2

u(x, t) = f(x-ct), where c is the speed of the wave. Negative sign means wave is right-travelling.

Question 3

What are mechanical waves? Give some examples.

Answer 3

Oscillation of matter. No mass transport, but energy transport through a medium. e.g. waves on a string, water waves, sound waves, seismic waves

Question 4

What are electromagnetic waves? Give some examples.

Answer 4

Waves which travel through things. e.g. light, ac electricity.

Question 5

What are matter waves

Answer 5

Probability amplitude for finding a particle at a given place.

Question 6

What is the general equation for a plane wave?

Answer 6

u(x,t) = A0 cos(kx-ωt+φ), where A0 is the amplitude, k the wavenumber, ω the angular frequency and phi the phase shift.

Question 7

What is the equation for k, wavenumber?

Answer 7

k = 2*pi/λ

Question 8

What is the equation for angular frequency ω?

Answer 8

ω = 2*pi/T

Question 9

What is the equation for a wave with fixed x-coordinate?

Answer 9

y = A0cos(φ-ωt) = A0cos(ωt-φ)

Question 10

How do you work out the speed of a wave in the y-direction?

Answer 10

dy/dt = -A0ω*sin(ωt-φ)

Question 11

What is the equation for phase velocity?

Answer 11

c = ω/k = vp

Question 12

How do you write the equation for a plane wave in the complex exponential form?

Answer 12

u(x, t) = Re(A0e^(i(kx-ωt+φ)))
Let A = A0e^iφ
u(x,t) = Re(Ae^(i(kx-ωt)))

Question 13

What is the wave equation?

Answer 13

1/v^2 * d^2 u/dt^2 = d^2 u/dx^2

Question 14

How do you write the wave equation in a three-dimensional generalisation?

Answer 14

1/v^2 * d^2 u/dt^2 = d^2 u/dx^2 + d^2 u/dy^2 + d^2u/dz^2 = ∇^2 u

Question 15

How is a plane wave a solution to the wave equation?

Answer 15

Take the complex form of the plane wave equation and put it through the wave equation.

Question 16

How do you calculate the net force on a string as a function of x and t?

Answer 16

Consider small element of the string. Horizontal force: T(x+dx)cos(theta) = T0
Vertical force: Fy(x+dx) = T(x+dx)sin(theta) = T0tan(theta) = T0dy/dx

Question 17

What is the net vertical force on element dx?

Answer 17

Fy(x+dx)-Fy(x) = dx dFy/dx

Question 18

What is the wave equation including mass per unit length μ?

Answer 18

μ/T0 * d^2 y/dt^2 = d^2 y/dx^2

Question 19

What is the equation for velocity of a wave involving tension T?

Answer 19

v = sqrt(T/μ)

Question 20

What is the equation for the power of a wave?

Answer 20

P(x,t) = -T*dy/dx *dy/dt

where d is partial

Question 21

What is the impedance of a wave?

Answer 21

Z = T/v

Question 22

What is the equation for the mean power of a wave?

Answer 22

P = 1/2 * Z * ω^2 * A0^2

Question 23

In 3D, what is the equation for intensity as a function of r?

Answer 23

I(r) = P/(4Ipi*r^2)