Standing Waves

Question 1

Wave Function for Standing Waves

Answer 1

y == 2A0 sin(kx) cos(ωt)

Question 2

Standing Waves

Description

Answer 2

• every part of the string vibrates with SHM with the same frequency and phase
• amplitude is dependent on x

Question 3

Standing Waves on a String

Answer 3

• natural patterns of oscillation of the string each with a different characteristic frequency
• the length of the string is always equal to an integer number of half wavelengths
• fundamental has the longest wavelength, λ/2=L

Question 4

Harmonics and Overtones

Answer 4

• the fundamental is the lowest frequency mode
• frequency of each normal mode is an integer multiple of the fundamental frequency
• these integer multiples are called harmonics
• the fundamental is the first harmonic and so on
• these normal mode frequencies are also referred to as overtones, where the 2nd harmonic is the first over tone, the 3rd harmonic is the 2nd overtone so on

Question 5

Standing Waves on a String

Description

Answer 5

when a stretched string is distorted so that the initial shape corresponds to a harmonic, only that particular normal mode is excited, so the string vibrates with the frequency of that harmonic

Question 6

Standing Waves – Musical Instruments

Description

Answer 6

-when a string is struck/plucked on a piano/guitar, its initial shape is no that of a single normal mode. Several normal modes are excited resulting in vibration that includes several harmonic frequencies. Typically, the fundamental has the largest amplitude

Question 7

Sound Waves

Answer 7

• longitudinal
• move through gases, liquids or solids
• speed depends on the properties of the medium

Question 8

Classifying Sound Waves

Answer 8

Audible Waves - 20-20000 Hz
Infrasonic Waves - frequency below audible
Ultrasonic Waves - frequency higher than audible

Question 9

Energy of a Sound Wave

Answer 9

Etot = 1/2 ρ(ωs0)² A vt

s0 = amplitude of sound wave
A = surface area
v = speed of wave

Question 10

Sound Wave - Wave FUnction

Answer 10

S = S0 cos(ωt)

Question 11

Maximum Velocity of a Sound Wave

Answer 11

Vmax = ωS0

Question 12

Energy Density of a Sound Wave

Answer 12

1/2 ρω²S0²

Question 13

Power

Definition (Sound Wave)

Answer 13

rate at which energy is transmitted to each layer

Question 14

Power of a Sound Wave

Answer 14

P = 1/2 ρ(ωs0)² A v

Question 15

Intensity of a Sound Wave

Answer 15

I = 1/2 ρ(ωs0)² v

Question 16

Standing Waves in Air Columns

Open at Both Ends

Answer 16

• nodes at each end of the tube
• first normal mode, L = λ/2
• second normal mode, L = λ
• third normal mode, L = 3λ/2
• the integer numbers of half wavelengths of the normal modes are equal to the length

Question 17

Standing Waves in Air Columns

Open at One End

Answer 17

• antinode at the closed end
• node at the open end
• first normal mode, L = λ/4
• second normal mode, L = 3λ/4
• third normal mode, L = 5λ/4
• for the normal modes L is equal to odd integer numbers of quarter wavelengths

Question 18

Pressure Wave

Velocity Equation

Answer 18

v = √(B/p)

B = bulk modulus of compressibility
p = density 
v = velocity

Question 19

Bulk Modulus of Compressibility

Equation

Answer 19

B = γp

γ = Cp/Cv

Question 20

Pressure Wave Equation

Answer 20

ΔP = BkS0 cos(kx-ωt)

Question 21

Amplitude of Sound Wave

Pressure Equation

Answer 21

S0 = ΔP0 / ρωv

Question 22

Sound Wave Intensity

Pressure Equation

Answer 22

I = 1/2 ρω²v {ΔP0/ρωv}²

Question 23

Spherical Waves

Answer 23

• in a uniform medium, waves move outwards from the source at constant speed
• from a point source (a small object oscillating with SHM) are produced with spherical wavelengths

Question 24

Intensity and Amplitude

Answer 24

I ∝ A²

-> I = cA²
A = √(I/c)

c = constant of proportionality

Question 25

Decibels

Answer 25

-a lotharithmic scale

β = 10 log(10){I/I0)

β = measured in decibels (dB)
I = Intensity of the sound
I0 = threshold of hearing