© 2013 by Fernando Caracena
The wave equation
Waves are everywhere. Look for them in the air. They are in the water. They have a mathematical structure that is distinct from the way particles move, which we explore here.
- Fig. 1. Wave clouds formed by the airflow over the mountains.
By looking at the motion of standing waves in a tight string, we came up with the following partial differential equation that describes the motion of the sting
∂2yn(x,t)/∂2x = (σ /T) ∂2yn(x,t)/∂t2. (I.10a)
That equation is a specialized form of a more general one, called the homogeneous wave equation in one dimension+ time,
∂2Φ(x,t)/∂x2-∂2Φ(x,t)/c2∂t2 =0, (1a)
which is sometimes written in differential operator form,
(∂2/∂x2-∂2/c2∂t2) Φ(x,t)=0. (1b)
The rule in differential operator format is that unless otherwise indicated the operators act only to the right on the functions that they are applied to. For example, a function multiplied on the left of (1b) would no be differentiated by the differential operator.
The wave equation in three dimensions and time is described by the following:
(2a)
Where the d'Alembertian operator is defined as
. (2b)
In one dimension, the solution to the wave equation (1a or 1b) is a combination of a wave propagating to the right, and one to the left,
Φ(x,t) = g(x-ct) + h(x+ct). (3a)
To show this differentiate twice with respect to x
∂2Φ(x,t)/∂x2= d2 g(ξ)/dξ2(dξ/dx)2+ d2 h(ρ))/dρ2 (dρ/dx)2 ,
which simplifies to
∂2Φ(x,t)/∂x2= d2 g(ξ)/dξ2 + d2 h(ρ))/dρ2 , (3b)
where
ξ = x-ct,
ρ = x+ct,
dξ/dx = 1,
and
dρ/dx= 1.
The time derivative part gives
∂2Φ(x,t)/∂2t2=c2d2 g(ξ)/dξ2 + c2d2 h(ρ))/dρ2
or
c-2∂2Φ(x,t)/∂2t2= d2 g(ξ)/dξ2 +d2 h(ρ))/dρ2 , (3c)
where
dξ/dt = c,
and
dρ/dt= c.
Entering (3b) and (3c) into (1a)
we see that the equation is identically satisfied
d2 g(ξ)/dξ2 + d2 h(ρ))/dρ2 -[d2 g(ξ)/dξ2 +d2 h(ρ))/dρ2] = 0
0 = 0.
Further, comparing (I.10a) and (1a), we can write an equation for the speed of waves in the tight string in terms of the tension and linear, mass density of the string,
c=√(T/σ) . (1c)
Harmonic wave form
The general for of a harmonic wave, which satifies the one-dimensional wave equation is
Φ(x,t) = a sin(k x - ω t) + b cos(k x - ω t), (4a)
where
k=2 π / λ ,
ω = 2 π ν ,
λ is the wavelength and ν, the frequency.
A more compact form of (4a) is
Φ(x,t) = A sin(k x - ω t + θ), (4b)
where θ is an initial phase angle.
Exercise, by using the trigonometric identity
sin(C+D)=sin(C)cos(D)+cos(C)sin(D) (4c)
show that
a= A cos(θ)
and
b=A sin(θ).
By plugging the wave form given by (4b) into (1a), we get
-(k2 - ω2 /c2) Φ(x,t) =0,
which is satisfied only if
(k2 - ω2 /c2) =0,
or
ω = k c. (4d)
2 π ν = 2 π c / λ.
Cancelling out the 2 π on both sides and multiplying through by λ results in a relationship between wavelength, frequency and the phase speed of the waves,
λ ν = c. 5)
Standing Waves
Fig. 2. Waves in a pond.
Here we will use the harmonic solutions to the wave equation to show that standing waves result from sine waves of the same wavelength streaming in opposite directions:
Φ(x,t) = 0.5 A [sin(k x - ω t)+sin(k x + ω t)]. (6a)
Use the trigonometric identity for the sum of two sine waves (4c) to rewrite (6a)
Φ(x,t) = 0.5 A [sin(k x)cos(ω t)-cos(k x)sin(ω t)+sin(k x)cos(ω t)+cos(k x)sin(ω t)]
or
Φ(x,t) = 0.5 A 2 sin(k x)cos(ω t)
Φ(x,t) = A sin(k x)cos(ω t). (7b)
Wrap up
Wave motion and the motion of particles are the two main concepts in describing motion in physics. Wave phenomena occur on a variety of levels in physics from ripples on the surface of ponds to electromagnetic waves in radio; yet, all follow the same general wave equation such as (2a) and a modified wave equation that contains a source term on the right instead of zero. In quantum mechanics, which deals with the behavior of atomic and subatomic particles, both the ideas of particles and waves are necessary for understanding that world.