Oscillations and Waves 2

Question 1

Speed of Waves on a String

Answer 1

v = √(Ft/μ)

Question 2

Speed of Sound Waves

Answer 2

v = √(B/ρ)

Question 3

Sound Waves in a Gas

Answer 3

v = √(γRT/M)

Question 4

The Wave Equation

Answer 4

∂²y/∂x² = 1/v² ∂²y/∂t²

Question 5

Harmonic Waves

Wave Function

Answer 5

y(x,t) = A sin(kx±ωt)
A = amplitude
k = wave number
ω = angular frequency
minus sign for a wave travelling in the positive x direction and a plus sign for a wave travelling in the negative x direction

Question 6

Wave Number

Answer 6

k = 2π / λ

Question 7

Angular Frequency

Answer 7

ω = 2πf = 2π/T

Question 8

Wave Speed

Answer 8

v = fλ = ω/k

Question 9

Energy in a Harmonic Wave

Answer 9

The energy in a harmonic wave is proportional to the square of the amplitude

Question 10

Power for Harmonic Waves on a String

Answer 10

Pav = 1/2 μvω²A²

Question 11

Harmonic Sound Waves

Answer 11

Sound waves can be considered to be either displacement waves or pressure waves. The human ear is sensitive to sound waves of frequencies from 20Hz-20000Hz. In a harmonic sound wave, the pressure and displacement are 90° out of phase.

Question 12

Harmonic Sound Waves

Amplitudes

Answer 12

Pressure and displacement amplitudes are related by
p0 = ρωvs0

p0 = pressure amplitude
s0 = displacement amplitude
ρ = density of the medium

Question 13

Harmonic Sound Waves

Energy Density

Answer 13

ηav = (ΔE)av / ΔV = 1/2 ρω²s0²

Question 14

Harmonic Sound Waves

Intensity

Answer 14

I = Pav / A
Average intensity of a sound wave:
I = ηav*v = 1/2 ρω²s0²v = 1/2 * p0²/ρv

Question 15

Harmonic Sound Waves

Intensity Level

Answer 15

β = (10 dB) log (I/I0)

where I0 = 10^(-12) W/m² is taken as the threshold of hearing

Question 16

Reflection and Refraction

Answer 16

When a wave is incident on a boundary surface that separates two regions of differing wave speed, part of the wave is transmitted and part is reflected.

Question 17

Reflection Coefficient

Answer 17

r = (v2-v1) / (v1+v2)

Question 18

Transmission Coefficient

Answer 18

τ = 2*v2/(v1+v2)

Question 19

Diffraction

Answer 19

If a wavefront is partially blocked by an obstacle, the unblocked part of the wavefront diffracts (bends) into the region behind the obstacle.

Question 20

Ray Approximation

Answer 20

If a wavefront is partially blocked by an obstacle, almost all of the diffraction occurs for that part of the wavaefront that passes within a few wavelengths of the edge.
For those part of the wavefront that pass farther from the edge than a few wavelengths, diffraction is negligible and the wave propagates in straight lines in the direction of the incident rays.

Question 21

Doppler Effect

Answer 21

When a sound source and receiver are in relative motion, the received frequency fr is higher than the frequency of the source fs if their separation is decreasing, and lower if their separation is increasing.

Question 22

Doppler Effect

Wavelength, Moving Source

Answer 22

λ = (v±us)/fs

Question 23

Doppler Effect

Frequency, Moving Receiver

Answer 23

fr = (v±ur)/λ

Question 24

Doppler Effect

Either Source or Receiver Moving

Answer 24

fr = (v±ur)/(v±us) * fs

Question 25

Doppler Effect

Small Speeds of Source or Receiver

Answer 25

Δf / fs ≈ ± u/v

where u = us ± ur

Question 26

Relativistic Doppler Shift

Answer 26

fr = √ [(c±u)/(c±u)] * fs

Question 27

Shock Waves

Answer 27

When the source speed is greater than the wave speed, the waves behind the source are confined to a cone of angle θ, the mach angle

Question 28

Mach Angle

Answer 28

sin θ = v/u

Question 29

Mach Number

Answer 29

Mach number = u/v