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Acoustics Formulas

Complete collection of acoustics formulas with detailed explanations.

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Sound Wave Properties

Sound Wave Speed

v=Bρv = \sqrt{\frac{B}{\rho}}

Sound speed equals square root of bulk modulus divided by density.

Notation:

v:Sound speed
B:Bulk modulus
ρ:Density

Units:

v:m/s
B:Pa
ρ:kg/m³

Applications:

  • •Ultrasound imaging
  • •Acoustic measurements
  • •Material characterization

Limitations:

Linear elastic media

Sound Intensity

I=PA=12ρω2A2vI = \frac{P}{A} = \frac{1}{2}\rho\omega^2A^2v

Sound intensity equals power per unit area or half density times angular frequency squared times amplitude squared times speed.

Notation:

I:Sound intensity
P:Power
A:Area
ρ:Density
ω:Angular frequency
A_amp:Amplitude
v:Sound speed

Units:

I:W/m²
P:W
A:m²
ρ:kg/m³
ω:rad/s
v:m/s

Applications:

  • •Noise measurement
  • •Acoustic design
  • •Hearing protection

Limitations:

Plane waves

Sound Pressure Level

SPL=20log⁡10(PP0)SPL = 20\log_{10}\left(\frac{P}{P_0}\right)

Sound pressure level equals 20 times logarithm base 10 of pressure ratio to reference pressure.

Notation:

SPL:Sound pressure level
P:Sound pressure
P₀:Reference pressure

Units:

SPL:dB
P:Pa
P₀:2 × 10⁻⁵ Pa

Applications:

  • •Noise assessment
  • •Environmental monitoring
  • •Audio engineering

Limitations:

Logarithmic scale

Wavelength-Frequency Relation

λ=vf\lambda = \frac{v}{f}

Wavelength equals sound speed divided by frequency.

Notation:

λ:Wavelength
v:Sound speed
f:Frequency

Units:

λ:m
v:m/s
f:Hz

Applications:

  • •Room acoustics
  • •Speaker design
  • •Wave interference

Limitations:

Single frequency waves

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Resonance and Standing Waves

Resonant Frequency

f=nv2Lf = \frac{nv}{2L}

Resonant frequency equals harmonic number times speed divided by twice length.

Notation:

f:Resonant frequency
n:Harmonic number
v:Sound speed
L:Length

Units:

f:Hz
n:Dimensionless
v:m/s
L:m

Applications:

  • •Musical instruments
  • •Acoustic cavities
  • •Pipe resonators

Limitations:

Closed-end pipes

Open Pipe Resonance

f=nv2Lf = \frac{nv}{2L}

Open pipe resonant frequency equals harmonic number times speed divided by twice length.

Notation:

f:Resonant frequency
n:Harmonic number
v:Sound speed
L:Length

Units:

f:Hz
n:Dimensionless
v:m/s
L:m

Applications:

  • •Wind instruments
  • •Organ pipes
  • •Acoustic tubes

Limitations:

Open-end pipes

String Resonance

f=n2LTμf = \frac{n}{2L}\sqrt{\frac{T}{\mu}}

String resonant frequency equals harmonic number times square root of tension divided by linear density, all divided by twice length.

Notation:

f:Resonant frequency
n:Harmonic number
T:Tension
μ:Linear density
L:Length

Units:

f:Hz
n:Dimensionless
T:N
μ:kg/m
L:m

Applications:

  • •String instruments
  • •Guitar tuning
  • •Vibrating strings

Limitations:

Fixed ends

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Doppler Effect

Doppler Effect (Moving Source)

f′=f1±vsvf' = \frac{f}{1 \pm \frac{v_s}{v}}

Observed frequency equals source frequency divided by one plus or minus source velocity divided by sound speed.

Notation:

f':Observed frequency
f:Source frequency
v_s:Source velocity
v:Sound speed

Units:

f':Hz
f:Hz
v_s:m/s
v:m/s

Applications:

  • •Siren frequency
  • •Moving sound sources
  • •Acoustic measurements

Limitations:

Source moving toward/away

Doppler Effect (Moving Observer)

f′=f(1±vov)f' = f\left(1 \pm \frac{v_o}{v}\right)

Observed frequency equals source frequency times one plus or minus observer velocity divided by sound speed.

Notation:

f':Observed frequency
f:Source frequency
v_o:Observer velocity
v:Sound speed

Units:

f':Hz
f:Hz
v_o:m/s
v:m/s

Applications:

  • •Moving observer
  • •Relative motion
  • •Frequency shift

Limitations:

Observer moving toward/away

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Acoustic Impedance

Acoustic Impedance

Z=ρvZ = \rho v

Acoustic impedance equals density times sound speed.

Notation:

Z:Acoustic impedance
ρ:Density
v:Sound speed

Units:

Z:Pa·s/m
ρ:kg/m³
v:m/s

Applications:

  • •Sound reflection
  • •Transmission loss
  • •Material properties

Limitations:

Normal incidence

Reflection Coefficient

R=Z2−Z1Z2+Z1R = \frac{Z_2 - Z_1}{Z_2 + Z_1}

Reflection coefficient equals impedance difference divided by impedance sum.

Notation:

R:Reflection coefficient
Z₁:Impedance of medium 1
Z₂:Impedance of medium 2

Units:

R:Dimensionless
Z₁, Z₂:Pa·s/m

Applications:

  • •Sound barriers
  • •Acoustic materials
  • •Echo analysis

Limitations:

Normal incidence

Transmission Coefficient

T=4Z1Z2(Z1+Z2)2T = \frac{4Z_1Z_2}{(Z_1 + Z_2)^2}

Transmission coefficient equals four times product of impedances divided by square of impedance sum.

Notation:

T:Transmission coefficient
Z₁:Impedance of medium 1
Z₂:Impedance of medium 2

Units:

T:Dimensionless
Z₁, Z₂:Pa·s/m

Applications:

  • •Sound transmission
  • •Acoustic design
  • •Noise control

Limitations:

Normal incidence