Module I – physics of sound
What is sound
Wave types:
Longitudinal wave → back and forth motion in
direction of the sound
Transverse wave → back and forth motion
perpendicular to the sound direction
Sound sources:
1. Vibrating bodies: displaces air and causes the local air pressure to increase and decrease
slightly, these fluctuations travel outwards as a sound wave.
2. Changing airflow: air flow increases or decreases resulting in a sound wave (vocal folds open
and close when speaking).
3. Time-dependent heat sources: explosion produces a bang due to the expansion of the air
caused by its rapid heating (same for thunder).
4. Supersonic flow: shock waves result when a supersonic airplane or a speeding rifle bullet
forces air to flow faster than the speed of sound
Physics
formula
Distance, speed and velocity 𝑣 = 𝑠𝑡 = 𝑑𝑠
𝑑𝑡
v= speed [m/s], s= distance/displacement [m], t=time [s]
𝑑𝑣
Force 𝐹 = 𝑚𝑎 = 𝑚
𝑑𝑡
F=force [N], a=acceleration [m/s2], m=mass [kg]
acceleration 𝑎𝑎𝑣 = ∆𝑣 = 𝑑𝑣
∆𝑡 𝑑𝑡
aav=average acceleration [m/s2], dv=change of speed [m/s], dt=change
of time [s]
F
Pressure p= ⊥
A
p=pressure [pa] or [N/m2], F⊥ = force perpendicular to surface [N],
A= area surface [m2]
Work (general) 𝑊 = 𝐹𝑠
W= work [J] or [N/m], F=force [N], s= distance/displacement [m]
Kinetic energy 𝑚𝑣 2
𝐸𝑘 = 𝐾𝐸 = 2
Ek= kinetic energy [J], m=mass [kg], v= speed [m/s]
Gravitational potential energy E𝑔 = E𝑝 = 𝑃𝐸 = 𝑚𝑔ℎ
Eg= potential energy or gravitational potential energy [j], m=mass [kg],
g= 9.81 [m/s2], h= height [m]
Potential energy spring 𝐾𝑦 2
E𝑝 = 𝑃𝐸 =
2
Ep= potential energy [j], K=spring constant [N/m], y= displacement
spring [m]
Potential energy (pressure 𝑉𝑝2
E𝑝 = 𝑃𝐸 =
2𝑃0
difference) Ep= potential energy [j], V= volume [m3], P0=atmospheric pressure [pa], p
= pressure [pa]
, 2
Potential energy (motion of a E𝑝 = 𝑃𝐸 = 2𝑇𝑦
𝐿
string) E = potential energy [j], T=tension of string [N], L=length [m],
p
Guitar string displaced from y=displacement from origin [m]
midpoint
Power 𝑃 = 𝑊𝑡
P=power [W] or [J/s], W= work [j], t=time [s]
Waves
Definition of wave: Disturbance in a medium that travels from one position to another position
Representation of wave in space and time (wave function)
𝜓(𝑥, 𝑡) = 𝐴 𝑠𝑖𝑛(𝜔𝑡 ± 𝜅𝑥)
𝜅 = wavenumber [m-1], 𝜔 = angular frequency [rad/s]
v = speed of sound [m/s]
2𝜋 1
𝜅= 𝜆
𝜔 = 2𝜋/𝑇 𝑓=𝑇 f = frequency [Hz]
𝜆 = wavelength [m]
Δ𝑥 1 𝜔 A = amplitude [m]
𝑣= Δ𝑡
= 𝜆 (𝑇) = 𝜆𝑓 = 𝜅
Speed of longitudinal waves in solid
𝐸
𝑣=√
𝜌
𝐸 = Elastic modulus [N], 𝜌 = density of solid [Kg/m3], v = speed of traveling wave [m/s]
Speed of transverse waves on a string
𝑇
𝑣 = √𝜇
𝑇 = Tension of the string [N], 𝜇 = mass per unit length [Kg/m]
Standing wave
Wave that oscillates in time but whose peak amplitude profile does not move in space →
combination of two waves: 1. Moving in opposite direction 2. Each wave has same A and f.
Exhibit phenomena
Interference
Interfering of two traveling waves in a medium → constructive (waves boost each other, in phase)
and destructive (waves cancel each other, out of phase) interference.
, Interference → causes maxima and minima in certain directions
Reflection
Sound wave reflect back upon hitting a surface → image source principle is
used
Refraction
Change in the speed of sound → change in direction of wave propagation or bending of the waves →
atmospheric effects, change of medium (air to glass)
Diffraction
Bending of waves around corners or trough aperture (hole, opening).
Amount of diffraction → size of aperture in relation to the wavelength
Opening larger then wavelength → little diffraction
Opening smaller or similar to wavelength → significant diffraction
Doppler effect
Apparent shift in frequency when source or receiver is moving
(𝑣±𝑣0 )
𝑓 = 𝑓0 𝑣±𝑣𝑠
f0 = f source [Hz], v = speed of sound [m/s], v0 =
velocity observer [m/s], vs = velocity of source [m/s]
mass spring systems
used for understanding resonance and simple harmonic motion
the restoring force of a mass spring system is proportional to the displacement y form its equilibrium
position.
𝐹 = −𝑘𝑦 → hooke’s law
𝐹 = 𝑚𝑎 → Newton’s 2nd law
Combining the equations yields:
𝑑𝑣 𝑑2 𝑦 𝑑2 𝑦 𝑘𝑦
−𝑘𝑦 = 𝑚𝑎 → 𝑚𝑎 + 𝑘𝑦 = 0 → 𝑚 + 𝑘𝑦 = 0 → 𝑚 + 𝑘𝑦 = 0 → + =0
𝑑𝑡 𝑑𝑡 2 𝑑𝑡 2 𝑚
The remaining equation is a second order linear X(t) = displacement as function of time
differential equation with solutions: A = amplitude of motion [m]
𝜔 = angular frequency = 2𝜋𝑓
𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑡 + 𝜙) or 𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝜔𝑡 + 𝜙) 𝜙 = phase constant (start point, and direction)
𝑘 t = time [s]
𝜔 = √𝑚
What is sound
Wave types:
Longitudinal wave → back and forth motion in
direction of the sound
Transverse wave → back and forth motion
perpendicular to the sound direction
Sound sources:
1. Vibrating bodies: displaces air and causes the local air pressure to increase and decrease
slightly, these fluctuations travel outwards as a sound wave.
2. Changing airflow: air flow increases or decreases resulting in a sound wave (vocal folds open
and close when speaking).
3. Time-dependent heat sources: explosion produces a bang due to the expansion of the air
caused by its rapid heating (same for thunder).
4. Supersonic flow: shock waves result when a supersonic airplane or a speeding rifle bullet
forces air to flow faster than the speed of sound
Physics
formula
Distance, speed and velocity 𝑣 = 𝑠𝑡 = 𝑑𝑠
𝑑𝑡
v= speed [m/s], s= distance/displacement [m], t=time [s]
𝑑𝑣
Force 𝐹 = 𝑚𝑎 = 𝑚
𝑑𝑡
F=force [N], a=acceleration [m/s2], m=mass [kg]
acceleration 𝑎𝑎𝑣 = ∆𝑣 = 𝑑𝑣
∆𝑡 𝑑𝑡
aav=average acceleration [m/s2], dv=change of speed [m/s], dt=change
of time [s]
F
Pressure p= ⊥
A
p=pressure [pa] or [N/m2], F⊥ = force perpendicular to surface [N],
A= area surface [m2]
Work (general) 𝑊 = 𝐹𝑠
W= work [J] or [N/m], F=force [N], s= distance/displacement [m]
Kinetic energy 𝑚𝑣 2
𝐸𝑘 = 𝐾𝐸 = 2
Ek= kinetic energy [J], m=mass [kg], v= speed [m/s]
Gravitational potential energy E𝑔 = E𝑝 = 𝑃𝐸 = 𝑚𝑔ℎ
Eg= potential energy or gravitational potential energy [j], m=mass [kg],
g= 9.81 [m/s2], h= height [m]
Potential energy spring 𝐾𝑦 2
E𝑝 = 𝑃𝐸 =
2
Ep= potential energy [j], K=spring constant [N/m], y= displacement
spring [m]
Potential energy (pressure 𝑉𝑝2
E𝑝 = 𝑃𝐸 =
2𝑃0
difference) Ep= potential energy [j], V= volume [m3], P0=atmospheric pressure [pa], p
= pressure [pa]
, 2
Potential energy (motion of a E𝑝 = 𝑃𝐸 = 2𝑇𝑦
𝐿
string) E = potential energy [j], T=tension of string [N], L=length [m],
p
Guitar string displaced from y=displacement from origin [m]
midpoint
Power 𝑃 = 𝑊𝑡
P=power [W] or [J/s], W= work [j], t=time [s]
Waves
Definition of wave: Disturbance in a medium that travels from one position to another position
Representation of wave in space and time (wave function)
𝜓(𝑥, 𝑡) = 𝐴 𝑠𝑖𝑛(𝜔𝑡 ± 𝜅𝑥)
𝜅 = wavenumber [m-1], 𝜔 = angular frequency [rad/s]
v = speed of sound [m/s]
2𝜋 1
𝜅= 𝜆
𝜔 = 2𝜋/𝑇 𝑓=𝑇 f = frequency [Hz]
𝜆 = wavelength [m]
Δ𝑥 1 𝜔 A = amplitude [m]
𝑣= Δ𝑡
= 𝜆 (𝑇) = 𝜆𝑓 = 𝜅
Speed of longitudinal waves in solid
𝐸
𝑣=√
𝜌
𝐸 = Elastic modulus [N], 𝜌 = density of solid [Kg/m3], v = speed of traveling wave [m/s]
Speed of transverse waves on a string
𝑇
𝑣 = √𝜇
𝑇 = Tension of the string [N], 𝜇 = mass per unit length [Kg/m]
Standing wave
Wave that oscillates in time but whose peak amplitude profile does not move in space →
combination of two waves: 1. Moving in opposite direction 2. Each wave has same A and f.
Exhibit phenomena
Interference
Interfering of two traveling waves in a medium → constructive (waves boost each other, in phase)
and destructive (waves cancel each other, out of phase) interference.
, Interference → causes maxima and minima in certain directions
Reflection
Sound wave reflect back upon hitting a surface → image source principle is
used
Refraction
Change in the speed of sound → change in direction of wave propagation or bending of the waves →
atmospheric effects, change of medium (air to glass)
Diffraction
Bending of waves around corners or trough aperture (hole, opening).
Amount of diffraction → size of aperture in relation to the wavelength
Opening larger then wavelength → little diffraction
Opening smaller or similar to wavelength → significant diffraction
Doppler effect
Apparent shift in frequency when source or receiver is moving
(𝑣±𝑣0 )
𝑓 = 𝑓0 𝑣±𝑣𝑠
f0 = f source [Hz], v = speed of sound [m/s], v0 =
velocity observer [m/s], vs = velocity of source [m/s]
mass spring systems
used for understanding resonance and simple harmonic motion
the restoring force of a mass spring system is proportional to the displacement y form its equilibrium
position.
𝐹 = −𝑘𝑦 → hooke’s law
𝐹 = 𝑚𝑎 → Newton’s 2nd law
Combining the equations yields:
𝑑𝑣 𝑑2 𝑦 𝑑2 𝑦 𝑘𝑦
−𝑘𝑦 = 𝑚𝑎 → 𝑚𝑎 + 𝑘𝑦 = 0 → 𝑚 + 𝑘𝑦 = 0 → 𝑚 + 𝑘𝑦 = 0 → + =0
𝑑𝑡 𝑑𝑡 2 𝑑𝑡 2 𝑚
The remaining equation is a second order linear X(t) = displacement as function of time
differential equation with solutions: A = amplitude of motion [m]
𝜔 = angular frequency = 2𝜋𝑓
𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑡 + 𝜙) or 𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝜔𝑡 + 𝜙) 𝜙 = phase constant (start point, and direction)
𝑘 t = time [s]
𝜔 = √𝑚
