Lead Article: Tables of Physics Formulae
This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Waves.
General Fundamental Quantites
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For transverse directions, the remaining cartesian unit vectors i and j can be used.
Quantity (Common Name/s)
(Common) Symbol/s
SI Units
Dimension
Number of Wave Cycles
N
{\displaystyle N\,\!}
dimensionless
dimensionless
(Transverse) Displacement
y
,
x
⊥
{\displaystyle y,x_{\bot }\,\!}
m
[L]
(Transverse) Displacement Amplitude
A
,
B
,
C
,
x
0
,
{\displaystyle A,B,C,x_{0},\,\!}
m
[L]
(Transverse) Velocity Amplitude
V
,
v
0
,
v
m
{\displaystyle V,v_{0},v_{\mathrm {m} }\,\!}
m s-1
[L][T]-1 (Transverse) Acceleration Amplitude
A
,
a
0
,
a
m
{\displaystyle A,a_{0},a_{\mathrm {m} }\,\!}
m s-2
[L][T]-2 (Longnitudinal) Displacement
x
,
x
∥
{\displaystyle x,x_{\parallel }\,\!}
m
[L]
Period
T
{\displaystyle T\,\!}
s
[T]
Wavelength
λ
{\displaystyle \lambda \,\!}
m
[L]
Phase Angle
δ
,
ϵ
,
ϕ
{\displaystyle \delta ,\epsilon ,\phi \,\!}
rad
dimensionless
General Derived Quantites
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The most general definition of (instantaneous) frequency is:
f
=
∂
N
∂
t
{\displaystyle f={\frac {\partial N}{\partial t}}\,\!}
For a monochromatic (one frequency) waveform the change reduces to the linear gradient:
f
=
Δ
N
Δ
t
{\displaystyle f={\frac {\Delta N}{\Delta t}}\,\!}
but common pratice is to set N = 1 cycle, then setting t = T = time period for 1 cycle gives the more useful definition:
f
=
1
T
{\displaystyle f={\frac {1}{T}}\,\!}
Quantity (Common Name/s)
(Common) Symbol/s
Defining Equation
SI Units
Dimension
(Transverse) Velocity
v
⊥
,
v
t
{\displaystyle v_{\bot },v_{\mathrm {t} }\,\!}
v
⊥
=
∂
x
⊥
∂
t
{\displaystyle v_{\bot }={\frac {\partial x_{\bot }}{\partial t}}\,\!}
m s-1
[L][T]-1 (Transverse) Acceleration
a
⊥
,
a
t
{\displaystyle a_{\bot },a_{\mathrm {t} }\,\!}
v
⊥
=
∂
v
⊥
∂
t
=
∂
2
x
⊥
∂
t
2
{\displaystyle v_{\bot }={\frac {\partial v_{\bot }}{\partial t}}={\frac {\partial ^{2}x_{\bot }}{\partial t^{2}}}\,\!}
m s-2
[L][T]-2 Path Length Difference
L
,
Δ
L
,
Δ
x
,
Δ
x
∥
{\displaystyle L,\Delta L,\Delta x,\Delta x_{\parallel }\,\!}
Δ
x
∥
=
x
∥
2
−
x
∥
1
{\displaystyle \Delta x_{\parallel }=x_{\parallel 2}-x_{\parallel 1}\,\!}
m
[L]
(Longnitudinal) Velocity
v
∥
,
v
p
{\displaystyle v_{\parallel },v_{\mathrm {p} }\,\!}
v
∥
=
Δ
x
∥
Δ
t
{\displaystyle v_{\parallel }={\frac {\Delta x_{\parallel }}{\Delta t}}\,\!}
m s-1
[L][T]-1 Frequency
f
,
ν
{\displaystyle f,\nu \,\!}
f
=
1
T
{\displaystyle f={\frac {1}{T}}\,\!}
Hz = s-1
[T]-1 Angular Frequency/ Pulsatance
ω
{\displaystyle \omega \,\!}
ω
=
2
π
f
=
2
π
/
T
{\displaystyle \omega =2\pi f=2\pi /T\,\!}
Hz = s-1
[T]-1 Time Delay, Time Lag/Lead
Δ
t
{\displaystyle \Delta t\,\!}
Δ
t
=
t
2
−
t
1
{\displaystyle \Delta t=t_{2}-t_{1}\,\!}
s
[T]
Scalar Wavenumber
k
{\displaystyle k\,\!}
Two definitions are used:
k
=
2
π
λ
{\displaystyle k={\frac {2\pi }{\lambda }}\,\!}
k
=
1
λ
{\displaystyle k={\frac {1}{\lambda }}\,\!}
In the formalism which follows, only the first
definition is used.
m-1
[L]-1 Vector Wavenumber
k
{\displaystyle \mathbf {k} \,\!}
Again two definitions are possible:
k
=
2
π
λ
x
^
∥
{\displaystyle \mathbf {k} ={\frac {2\pi }{\lambda }}\mathbf {\hat {x}} _{\parallel }\,\!}
k
=
1
λ
x
^
∥
{\displaystyle \mathbf {k} ={\frac {1}{\lambda }}\mathbf {\hat {x}} _{\parallel }\,\!}
In the formalism which follows, only the first
definition is used.
m-1
[L]-1 Phase Differance
Δ
ϵ
,
Δ
ϕ
,
δ
{\displaystyle \Delta \epsilon ,\Delta \phi ,\delta \,\!}
Δ
ϕ
=
ϕ
2
−
ϕ
1
{\displaystyle \Delta \phi =\phi _{2}-\phi _{1}\,\!}
rad
dimensionless
Phase
Φ
{\displaystyle \Phi \,\!}
(No standard symbol,
Φ
{\displaystyle \Phi \,\!}
is used
only here for clarity of equivalances )
Φ
=
x
−
v
t
+
Δ
x
N
=
λ
{\displaystyle \Phi ={\frac {x-vt+\Delta x}{N}}=\lambda \,\!}
Φ
=
k
x
−
ω
t
+
ϕ
=
2
π
N
{\displaystyle \Phi =kx-\omega t+\phi =2\pi N\,\!}
rad
dimensionless
Wave Energy
E
J
[M] [L]2 [T]-2 Wave Power
P
P
=
∂
2
E
∂
t
{\displaystyle P={\frac {\partial ^{2}E}{\partial t}}\,\!}
W = J s-1
[M] [L]2 [T]-3 Wave Intensity
I
I
=
∂
P
∂
A
{\displaystyle I={\frac {\partial P}{\partial A}}\,\!}
W m-2
[M] [T]-3 Wave Intensity (per unit Solid Angle)
I
I
=
∂
2
P
∂
Ω
∂
A
{\displaystyle I={\frac {\partial ^{2}P}{\partial \Omega \partial A}}\,\!}
Often reduces to
I
=
P
0
Ω
r
2
{\displaystyle I={\frac {P_{0}}{\Omega r^{2}}}\,\!}
W m-2 sr-1
[M] [T]-3
Phase
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point. Physically;
wave popagation in +x direction
x
⊥
>
0
⇒
ω
<
0
{\displaystyle x_{\bot }>0\Rightarrow \omega <0\,\!}
wave popagation in -x direction
x
⊥
<
0
⇒
ω
>
0
{\displaystyle x_{\bot }<0\Rightarrow \omega >0\,\!}
Phase angle can lag if:
ϕ
>
0
{\displaystyle \phi >0\,\!}
or lead if:
ϕ
<
0
{\displaystyle \phi <0\,\!}
Relation between quantities of space, time, and angle analogues used to describe the phase
Φ
{\displaystyle \Phi \,\!}
is summarized simply:
Δ
x
λ
=
Δ
t
T
=
ϕ
2
π
=
N
{\displaystyle {\frac {\Delta x}{\lambda }}={\frac {\Delta t}{T}}={\frac {\phi }{2\pi }}=N\,\!}
Standing Waves
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Harmonic Number
n
∈
Z
{\displaystyle n\in \mathbf {Z} \,\!}
Harmonic Series
f
n
=
v
λ
n
=
n
v
2
L
{\displaystyle f_{n}={\frac {v}{\lambda _{n}}}={\frac {nv}{2L}}\,\!}
Propagating Waves
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Wave Equation
Any wavefunction of the form
y
=
y
(
x
−
v
∥
t
)
{\displaystyle y=y\left(x-v_{\parallel }t\right)\,\!}
satisfies the hyperbolic PDE :
∇
2
y
=
1
v
∥
2
∂
2
y
∂
t
2
{\displaystyle \nabla ^{2}y={\frac {1}{v_{\parallel }^{2}}}{\frac {\partial ^{2}y}{\partial t^{2}}}\,\!}
Principle of Superposition for Waves
y
n
e
t
=
∑
i
(
y
i
)
{\displaystyle y_{\mathrm {net} }=\sum _{i}\left(y_{i}\right)\,\!}
General Mechanical Wave Results
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Average Wave Power
⟨
P
⟩
=
μ
v
ω
2
x
m
2
/
2
{\displaystyle \langle P\rangle =\mu v\omega ^{2}x_{m}^{2}/2\,\!}
Intensity
I
=
1
2
ρ
v
ω
2
s
0
2
{\displaystyle I={\frac {1}{2}}\rho v\omega ^{2}s_{0}^{2}\,\!}
Sound Waves
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Sound Intensity and Level
Quantity (Common Name/s)
(Common) Symbol/s
Sound Level
β
=
(
d
B
)
10
log
|
I
I
0
|
{\displaystyle \beta =\left(\mathrm {dB} \right)10\log \left|{\frac {I}{I_{0}}}\right|\,\!}
Sound Beats and Standing Waves
pipe, two open ends
f
=
v
/
λ
=
n
v
2
L
{\displaystyle f=v/\lambda ={\frac {nv}{2L}}\,\!}
Pipe, one open end
f
=
v
/
λ
=
n
v
4
L
{\displaystyle f=v/\lambda ={\frac {nv}{4L}}\,\!}
for n odd
Acoustic Beat Frequency
f
b
e
a
t
=
f
1
−
f
2
{\displaystyle f_{\mathrm {beat} }=f_{1}-f_{2}\,\!}
Sonic Doppler Effect
Sonic Doppler Effect
f
′
=
f
(
v
±
v
D
v
∓
v
S
)
{\displaystyle f'=f\left({\frac {v\pm v_{D}}{v\mp v_{S}}}\right)\,\!}
λ
=
λ
′
(
v
±
v
D
v
∓
v
S
)
{\displaystyle \lambda =\lambda '\left({\frac {v\pm v_{D}}{v\mp v_{S}}}\right)\,\!}
Mach Cone Angle
(Supersonic Shockwave, Sonic boom)
sin
θ
=
v
v
s
{\displaystyle \sin \theta ={\frac {v}{v_{s}}}\,\!}
Sound Wavefunctions
Acoustic Pressure Amplitude
Δ
p
0
=
v
ρ
ω
s
0
{\displaystyle \Delta p_{0}=v\rho \omega s_{0}\,\!}
Sound Displacement Function
s
=
s
0
cos
(
k
y
−
ω
t
)
{\displaystyle s=s_{0}\cos(ky-\omega t)\,\!}
Sound pressure-variation function
p
=
p
0
sin
(
k
y
−
ω
t
)
{\displaystyle p=p_{0}\sin(ky-\omega t)\,\!}
Superposition, Interferance/Diffraction
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Resonance
ω
d
=
ω
0
{\displaystyle \omega _{d}=\omega _{0}\,\!}
Phase and Interference
δ
2
π
=
λ
Δ
x
{\displaystyle {\frac {\delta }{2\pi }}={\frac {\lambda }{\Delta x}}\,\!}
Constructive Interference
λ
Δ
x
=
n
{\displaystyle {\frac {\lambda }{\Delta x}}=n\,\!}
Destructive Interference
λ
Δ
x
=
n
+
1
2
{\displaystyle {\frac {\lambda }{\Delta x}}=n+{\frac {1}{2}}\,\!}
n is any integer;
n
∈
Z
{\displaystyle n\in \mathbf {Z} \,\!}
Phase Velocities in Various Media
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The general equation for the phase velocity of any wave is (equivalent to the simple "speed-distance-time" relation, using wave quantities):
v
=
λ
f
=
ω
k
{\displaystyle v=\lambda f={\frac {\omega }{k}}\,\!}
The general equation for the group velocity of any wave is:
v
g
=
∂
ω
∂
k
{\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}\,}
A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media.
In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency.
The phase velocity is the rate at which the phase of the wave propagates in space.
The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.
Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function below called the Dispersion Relation , given in explicit form and implicit form respectively.
D
(
ω
,
k
)
=
0
{\displaystyle D\left(\omega ,k\right)=0}
ω
=
ω
(
k
)
{\displaystyle \omega =\omega \left(k\right)}
The use of ω (k ) for explicit form is standard, since the phase velocity ω /k and the group velocity dω /dk usually have convenient representations by this function.
For more specific media through which waves propagate, phase velocities are tabulated below. All cases are idealized, and the media are non-dispersive, so the group and phase velocity are equal.
Taut String
v
=
F
t
μ
{\displaystyle v={\sqrt {\frac {F_{\mathrm {t} }}{\mu }}}\,\!}
Solid Rods
v
=
Y
ρ
{\displaystyle v={\sqrt {\frac {Y}{\rho }}}\,\!}
Fluids
v
=
B
ρ
{\displaystyle v={\sqrt {\frac {B}{\rho }}}\,\!}
Gases
v
=
γ
R
T
M
m
=
γ
p
ρ
{\displaystyle v={\sqrt {\frac {\gamma RT}{M_{m}}}}={\sqrt {\frac {\gamma p}{\rho }}}\,\!}
The generalization for these formulae is for any type of stress or pressure p , volume mass density ρ , tension force F , linear mass density μ for a given medium:
v
=
p
ρ
=
F
μ
{\displaystyle v={\sqrt {\frac {p}{\rho }}}={\sqrt {\frac {F}{\mu }}}\,\!}
Pulsatances of Common Osscilators
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Sinusiodal Waves
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Equation of a Sinusiodal Wave is
y
=
A
sin
(
k
x
−
ω
t
+
ϕ
)
{\displaystyle y=A\sin \left(kx-\omega t+\phi \right)\,\!}
Recall that wave propagation is in
±
x
{\displaystyle \pm x\,\!}
direction for
∓
ω
{\displaystyle \mp \omega \,\!}
.
Sinusiodal waves are important since any waveform can be created by applying the principle of superposition to sinusoidal waves of varying frequencies, amplitudes and phases. The physical concept is easily manipulated by application of Fourier Transforms.
Wave Energy
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General Wavefunctions
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Sinusiodal Solutions to the Wave Equation
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The following may be duduced by applying the principle of superposition to two sinusiodal waves, using trigonometric identities. Most often the angle addition and sum-to-product formulae are useful; in more advanced work complex numbers and Fourier series and transforms are often used.
Wavefunction
Nomenclature
Superposition
Resultant
Standing Wave
y
1
+
y
2
=
A
sin
(
k
x
−
ω
t
)
{\displaystyle y_{1}+y_{2}=A\sin \left(kx-\omega t\right)\,\!}
+
A
sin
(
k
x
+
ω
t
)
{\displaystyle +A\sin \left(kx+\omega t\right)\,\!}
y
=
A
sin
(
k
x
)
cos
(
ω
t
)
{\displaystyle y=A\sin \left(kx\right)\cos \left(\omega t\right)\,\!}
Beats
⟨
ω
⟩
=
ω
1
+
ω
2
2
{\displaystyle \langle \omega \rangle ={\frac {\omega _{1}+\omega _{2}}{2}}\,\!}
⟨
k
⟩
=
k
1
+
k
2
2
{\displaystyle \langle k\rangle ={\frac {k_{1}+k_{2}}{2}}\,\!}
Δ
ω
2
=
ω
1
−
ω
2
2
{\displaystyle {\frac {\Delta \omega }{2}}={\frac {\omega _{1}-\omega _{2}}{2}}\,\!}
Δ
k
2
=
k
1
−
k
2
2
{\displaystyle {\frac {\Delta k}{2}}={\frac {k_{1}-k_{2}}{2}}\,\!}
y
1
+
y
2
=
A
sin
(
k
1
x
−
ω
1
t
)
{\displaystyle y_{1}+y_{2}=A\sin \left(k_{1}x-\omega _{1}t\right)\,\!}
+
A
sin
(
k
2
x
+
ω
2
t
)
{\displaystyle +A\sin \left(k_{2}x+\omega _{2}t\right)\,\!}
y
=
2
A
sin
(
⟨
k
⟩
x
−
⟨
ω
⟩
t
)
cos
(
Δ
k
2
x
−
Δ
ω
2
t
)
{\displaystyle y=2A\sin \left(\langle k\rangle x-\langle \omega \rangle t\right)\cos \left({\frac {\Delta k}{2}}x-{\frac {\Delta \omega }{2}}t\right)\,\!}
Coherant Interferance
y
1
+
y
2
=
A
sin
(
k
x
−
ω
t
)
{\displaystyle y_{1}+y_{2}=A\sin \left(kx-\omega t\right)\,\!}
+
A
sin
(
k
x
+
ω
t
+
ϕ
)
{\displaystyle +A\sin \left(kx+\omega t+\phi \right)\,\!}
y
=
2
A
cos
(
ϕ
2
)
sin
(
k
x
−
ω
t
+
ϕ
2
)
{\displaystyle y=2A\cos \left({\frac {\phi }{2}}\right)\sin \left(kx-\omega t+{\frac {\phi }{2}}\right)\,\!}
Note: When adding two wavefunctions togther the following trigonometric identity proves very useful:
sin
A
±
sin
B
=
2
A
sin
(
A
±
B
2
)
cos
(
A
∓
B
2
)
{\displaystyle \sin A\pm \sin B=2A\sin \left({\frac {A\pm B}{2}}\right)\cos \left({\frac {A\mp B}{2}}\right)\,\!}
Non-Solutions to the Wave Equation
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Exponentially Damped Waveform
y
=
A
e
−
b
t
sin
(
k
x
−
ω
t
+
ϕ
)
{\displaystyle y=Ae^{-bt}\sin \left(kx-\omega t+\phi \right)\,\!}
Solitary Wave
Common Waveforms
Triangular
Square
Saw-Tooth
External links
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