Lead Article: Tables of Physics Formulae
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}}\,\!}
f
=
Δ
N
Δ
t
{\displaystyle f={\frac {\Delta N}{\Delta t}}\,\!}
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 \,\!}
Φ
{\displaystyle \Phi \,\!}
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}}\,\!}
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\,\!}
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\,\!}
Φ
{\displaystyle \Phi \,\!}
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
y
=
y
(
x
−
v
∥
t
)
{\displaystyle y=y\left(x-v_{\parallel }t\right)\,\!}
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}}\,\!}
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}}\,\!}
λ
Δ
x
=
n
{\displaystyle {\frac {\lambda }{\Delta x}}=n\,\!}
λ
Δ
x
=
n
+
1
2
{\displaystyle {\frac {\lambda }{\Delta x}}=n+{\frac {1}{2}}\,\!}
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}}\,\!}
group velocity of any wave is:
v
g
=
∂
ω
∂
k
{\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}\,}
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)}
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.
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 }}}\,\!}
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)\,\!}
±
x
{\displaystyle \pm x\,\!}
∓
ω
{\displaystyle \mp \omega \,\!}
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)\,\!}
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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