Waves
Mark Robert Baker
Lecture 28
Contents
1 Waves
1
2 Mechanical Waves
2
3 Periodic Waves
3.1 Wave Speed . . . . . . . . . . . . . . . . . . . . . . .
4
5
4 The Wave Equation
4.1 The Wave Function . . . . . . . . . . . . . . . . . . .
4.2 Plotting The Wave Function . . . . . . . . . . . . . .
4.3 Phase Velocity . . . . . . . . . . . . . . . . . . . . . .
4.4 Particle Velocity and Acceleration . . . . . . . . . . .
4.5 Curvature in the sting, wave equation . . . . . . . . .
4.6 Solution to the Wave Equation . . . . . . . . . . . .
5
5
6
7
8
9
10
5 Basic Wave Properties/ Definitions
12
6 Energy in Wave Motion
13
1
Waves
Earlier in the course, we studied oscillations, in particular, simple
harmonic motion, where a particle oscillates back and forth indefinitely. One of the main reasons we study this in a first year physics
course is how (relatively) straightforward the equations are, yet how
clearly they show some of the basic features of a dynamic equation
1
of motion in physics. Another common type of motion that also has
a dynamic equation of motion which we can determine an analytic
solution for is one which we will study next: waves.
Wave motion is different from oscillations in the sense that the
waves travel through space, whereas oscillations oscillate back and
forth about a particular point. However, the equation of motion
and solution are similar, and oscillations appear in wave motion,
for example a period wave travelling on a string has at each point a
string particle oscillating back and forth in simple harmonic motion.
We will start by introducing the major types of waves which we
will consider in this course: mechanical waves.
2
Mechanical Waves
In this course we will primarily study mechanical waves. These
waves, unlike e.g. electromagnetic waves, travel within some material/ medium, such as on a string, in water, or in air. There are
three types of waves which we will consider:
1. Transverse waves (e.g. waves on a string)
2. Longitudinal waves (e.g. waves in a fluid)
3. Both 1. and 2. (e.g. waves on a fluid)
2
Figure 1: The three types of wave motion which we will consider.
Transverse waves
Transverse waves, such as a wave on a string, are ones where the
string moves transversely (that is, perpendicularly) to the motion
of the waves. For a wave on a string, suppose the wave is moving
in the +x direction, and the amplitude is in the y direction. The
string itself is moving up and down in the y direction, which is
perpendicular to the wave moving in x.
Longitudinal waves
Longitudinal waves, such as a wave within a fluid, are ones where the
medium (in this case, fluid particles) move parallel to the direction
of the wave. Therefore if the wave in the fluid is moving in the +x
direction, the fluid particular are also moving back and forth in the
x direction.
Both 1. and 2. waves
Waves that are neither transverse, nor longitudinal, are some combination of both types of motion. Therefore if a wave is travelling
in the +x direction, the particles are moving in e.g. a circle, not in
any distinct parallel or perpendicular direction relative to the wave
3
motion. An example of this type of wave is the waves on the surface
of a fluid.
3
Periodic Waves
For the most part we will consider periodic waves in this course.
For example, a wave on a string, or in a fluid, generated by a simple harmonic motion (SHM) oscillator. These waves are sinusoidal
waves, where each particle (e.g. on the string) is in SHM.
Figure 2: A periodic wave generated on a string.
Figure 3: A periodic wave generated in a fluid.
4
3.1
Wave Speed
The speed of a wave v is the speed of propagation for the wave. This
is how fast a given crest (peak) or trough in the wave is travelling,
for example if a wave is moving in the +x direction then this is how
fast the wave moves in this direction.
For periodic waves, all crests/ troughs of the wave move at the
same speed v. For periodic waves we can calculate this value by
multiplying the wave length λ by the frequency f ,
v = λf
4
(1)
The Wave Equation
One of the most, if not the most, important equations in physics is
the wave equation. It can be found in numerous areas of physics, for
quantities such as waves on a string, or even electromagnetic waves.
We will consider for now just the wave equation which represents
a wave on a string moving in one spatial dimension, for example a
wave travelling in x which is oscillating in the y direction. For such
a phenomena, the equation of motion which represents this wave is
known as the wave equation:
∂ 2 y(x, t)
1 ∂ 2 y(x, t)
=
(2)
∂x2
v 2 ∂t2
where v is the speed of the wave. The wave equation has the
solution y(x, t), which is a multivariable function, known as the
wave function. The above wave equation is often referred to as
a “1D Wave Equation” because the wave is travelling in 1 spatial
dimension.
4.1
The Wave Function
The wave function is the solution to the wave equation. In the
case of the wave equation presented above, the solution to the wave
equation is,
y(x, t) = A cos (κx − ωt + ϕ)
5
(3)
where the negative sign above indicates that the wave is travelling in the +x direction. If there was a positive sign there, it would
indicate that the wave is travelling in the -x direction.
The κ is known as the wave number,
2π
λ
(4)
ω = 2πf
(5)
κ=
and ω is the angular frequency,
which we will see can also be expressed for a periodic wave as,
ω = vκ
(6)
note that the amplitude of the wave A is independent of the
values for wave length λ by the frequency f .
4.2
Plotting The Wave Function
We can plot the wave function, however since it is a multivariable
function, if we wish to plot on a 2D plot we must hold one of the
variables constant. For example if we fix a point in time t, we can
plot y vs x,
Figure 4: A wave function plotted for a fixed time t.
6
here we can graphically interpret the wavelength of the wave.
This gives us an extended view of what the wave looks like at a
fixed point in time.
We could also fix a point in space x, and plot y vs t,
Figure 5: A wave function plotted for a fixed time x.
here we can graphically interpret a period of the wave. Since we
are at a fixed x, here we see the oscillation of one point on the string
as a function of time.
4.3
Phase Velocity
The angular frequency equation,
ω = vκ
(7)
is in fact a relationship for velocity known as the phase velocity,
ω
(8)
κ
this is the travelling velocity of a particular point on the wave
which we can find from the wave function,
v=
y(x, t) = A cos (κx − ωt + ϕ)
(9)
in order to travel with a fixed point on the wave, the value of cos
must be a constant, thus the argument of the cosine function must
be a constant,
7
κx − ωt + ϕ = constant
(10)
if we take the time derivative of both sides of this equation, ϕ
and the constant on the RHS are constants, thus are zero and,
κv − ω = 0
(11)
rearranging we get the phase velocity,
v=
4.4
ω
κ
(12)
Particle Velocity and Acceleration
We can determine the transverse velocity and acceleration of a particle on the string,
Figure 6: The transverse velocity and acceleration of a wave on a string are
those which are perpendicular to the motion of the wave (e.g. in x), that is the
velocity and acceleration in y of a point on the string.
to do this we have to start from the wave function,
y(x, t) = A cos (κx − ωt + ϕ)
(13)
and take the first and second time derivatives, we have the transverse velocity and acceleration:
∂y(x, t)
∂ 2 y(x, t)
, ay =
(14)
∂t
∂t2
These are partial derivatives, which mean we differentiate with
respect to t, and treat other input variable(s) (in this case just x) as a
vy =
8
constant. Therefore for our wave function y(x, t) = A cos (κx − ωt + ϕ),
when we take the partial derivative with respect to t, the combination of κx is treated as some constant. Differentiating we have:
vy =
∂y(x, t)
= Aω sin (κx − ωt + ϕ)
∂t
(15)
∂ 2 y(x, t)
ay =
= −Aω 2 cos (κx − ωt + ϕ)
(16)
2
∂t
the transverse acceleration can be expressed in terms of the wave
function as,
ay = −ω 2 y(x, t)
4.5
(17)
Curvature in the sting, wave equation
The curvature can be found by taking two position derivatives of
the wave function,
∂ 2 y(x, t)
= −Aκ2 cos (κx − ωt + ϕ)
∂x2
which can be expressed in terms of the wave function as,
∂ 2 y(x, t)
= −κ2 y(x, t)
∂x2
comparing this to the tangential acceleration,
∂ 2 y(x, t)
= −ω 2 y(x, t)
∂t2
we have the relationship,
(18)
(19)
(20)
1 ∂ 2 y(x, t)
1 ∂ 2 y(x, t)
=
κ2 ∂x2
ω 2 ∂t2
(21)
∂ 2 y(x, t)
κ2 ∂ 2 y(x, t)
=
∂x2
ω 2 ∂t2
(22)
∂ 2 y(x, t)
1 ∂ 2 y(x, t)
= 2
(23)
∂x2
v
∂t2
which is exactly the wave equation! Therefore we can recover the
wave equation by starting from the wave function and comparing
9
the transverse acceleration to the curvature in the string. This also
shows that the wave function is a solution to the wave equation,
because if we take two x derivatives and two t derivatives of the
wave function, the above relationship indeed holds.
4.6
Solution to the Wave Equation
We will briefly show that the function is a solution the wave equation
here. The wave equation (for waves travelling in x and y) is,
1 ∂ 2y
∂ 2y
=
(24)
∂x2
v 2 ∂t2
the v is the velocity of the wave. This velocity depends on the
medium the wave is travelling in (i.e. fluids, solids, gases), which
q
will be discussed later in the course. For example, in fluids v =
where B is the bulk modulus and ρ is the density of the fluid.
B
,
ρ
Just like in simple harmonic motion, the wave equation above
can be solved for a function which will represent the trajectory of
the wave. Once again, this function is a cos function,
y(x, t) = A cos (κx ± ωt + ϕ)
(25)
A couple important things to note here. First, ω is the angular
frequency of the wave [units radians per second],
ω = vκ
(26)
The curly K (the greek letter Kappa) represents the wave number
[units radians per metre],
2π
(27)
λ
where λ is the wavelength (the linear length of one complete cycle
along the axis). Again ϕ represents a phase shift. All of these constants are properties of a given wave. By knowing these constants
we define a specific wave trajectory (each wave we discuss will have
a specific set of associated constants, sometimes referred to as parameters of the wave model, that encode important properties such
κ=
10
as wavelength, velocity, amplitude etc.).
The ± depends on the direction the wave is travelling. If the
wave is travelling to the right, there is a − sign. If the wave is travelling to the left, there is a + sign.
We can now test our solution in the wave equation to verify that
it is a solution. First we should note the difference between the
derivative d we are used to, and the new curly ∂ used in this equation. The curly ∂ represents partial differentiation. For example,
on the right hand side, this mean to differentiate with respect to t
while holding all other input variables constant. This is necessary
because our solution y(x, t) is a function of both position x and time
t. Therefore on the right hand side, the x in the function will be
treated like one of the other constants. Substituting into both sides
a wave travelling to the right (− sign),
y(x, t) = A cos (κx − ωt + ϕ)
∂2
1 ∂2
A
cos
(κx
−
ωt
+
ϕ)
=
A cos (κx − ωt + ϕ)
∂x2
v 2 ∂t2
(28)
(29)
differentiating once on each side,
−κA
∂
1 ∂
sin (κx − ωt + ϕ) = ωA 2 sin (κx − ωt + ϕ)
∂x
v ∂t
(30)
and once more,
1
−κ2 A cos (κx − ωt + ϕ) = −ω 2 A 2 cos (κx − ωt + ϕ)
v
(31)
recall that we can relate the angular frequency to the wave number via ω = vκ. Inserting this expression to the right hand side,
−κ2 A cos (κx − ωt + ϕ) = −κ2 A cos (κx − ωt + ϕ)
(32)
Since the left hand side equals the right hand side, the solution
indeed satisfies the wave equation.
11
Basically what happened here is that the ω 2 and κ2 coefficients
that popped out from differentiation became the coefficient of the
wave equation:
1
κ2
=
v2
ω2
Allowed this solution to satisfy the equation of motion.
5
(33)
Basic Wave Properties/ Definitions
We have already mentioned a few of the parameters we will be using
to discuss waves, however it would be nice to have them all in one
compact location. In this section we will define properties of travelling waves that we will often be referring to. Of course we have the
wave equation,
∂ 2y
1 ∂ 2y
=
(34)
∂x2
v 2 ∂t2
where v is the velocity of the wave. We will most often be referred
to the solution in the sinusoidal form,
y(x, t) = A cos (κx ± ωt + ϕ)
(35)
The ± depends on the direction the wave is travelling. If the
wave is travelling to the right, there is a − sign. If the wave is travelling to the left, there is a + sign. This is because a negative ωt
phase means that our cos function is shifted slightly to the right (in
the positive x direction). Similarly a positive value here will shift
our cos function to the left, meaning our wave moves in the negative
direction for the + option as t increases.
Defining the parameters in the solution, first, ω is the angular
frequency of the wave [units radians per second],
ω = vκ
(36)
The curly K (the greek letter Kappa) represents the wave number
[units radians per metre],
κ=
2π
λ
12
(37)
where λ is the wavelength (the linear length of one complete cycle
along the axis). Again ϕ represents a phase shift. Since ω = 2πf ,
using the above two equations,
2πf = vκ
2π
λ
we have the relation for the speed of our wave,
(38)
2πf = v
(39)
v = fλ
(40)
In general, the wavelength, frequency and velocity of our wave
are the key parameters that describe a unique wave - knowing these
allows us to find the solution and the specific wave equation (model).
Don’t forget that the frequency f [s−1 ] is related to the period,
1
(41)
T
Using these expressions we can find many different relations between our parameters, such as,
f=
ω = vκ = 2πf =
2πv
2π
=
λ
T
(42)
etc. etc.
6
Energy in Wave Motion
We will briefly discuss energy in wave motion, in particular, energy
transported by a wave on a string. The energy transported by a
wave on a string is given by the following function:
p
(43)
P (x, t) = µT ω 2 A2 sin2 (κx − ωt)
where µ is the linear mass density of the string (mass per unit
length) and T is the tension. Plotting this function we have:
13
Figure 7: A plot of the power function, the maximum power, and the average
power.
This power function, in SI units Watts [W], tells us the energy
transport per second at a given position and time on the string. We
are often interested in two of the associated quantities, the maximum
power at any point in time, and the average power.
Maximum Power
The maximum power occurs when sin2 (κx − ωt) = 1 in the power
function. When this is true, we get the maximum power value.
Starting from the power function,
p
P (x, t) = µT ω 2 A2 sin2 (κx − ωt)
(44)
applying sin2 (κx − ωt) = 1 we get the maximum power Pmax ,
p
Pmax = µT ω 2 A2
(45)
Average Power
The average power over one full cycle can also be found from the
power function,
p
P (x, t) = µT ω 2 A2 sin2 (κx − ωt)
(46)
here we need to use the result that the average value of the sin
squared function over one full cycle is equal to 21 ,
14
1
2
therefore the average power over one full cycle is,
< sin2 (κx − ωt) >=
(47)
1p
µT ω 2 A2
(48)
2
We can write this (or any of the power equations for the waves
on a string) in alternate forms using the speed equation,
s
T
v=
(49)
µ
Pav =
for example we can replace µ in the average power equation,
1T 2 2
ω A
2v
or we can replace T in the average power equation,
Pav =
(50)
1
Pav = µvω 2 A2
(51)
2
therefore based on the information which is known, we can find
the power for different pairs of the variables T , µ and v.
Wave Intensity
A final note in this section is on wave intensity. For waves such as
sound waves which are transported out in 3D, the average intensity
I through a sphere radius r surrounding the source is,
P
(52)
4πr2
therefore intensity decreases the further we get from the source.
Wave intensity is important in the study of e.g. electromagnetic
waves.
I=
15