Unit 1 WAVES by lindahy

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									 Unit 1

  WAVES ............................................................................................... 3
   Definitions......................................................................................... 3
   Speed of waves in various media:..................................................... 9
   Reflection, transmission and absorption ......................................... 12
   Energy (power) carried by waves ................................................... 15
   Interference of waves ...................................................................... 16
   Standing waves................................................................................ 21
   Resonance ....................................................................................... 22

  Unit 1 Waves                                                                                              2
 Why study waves?

 •   Everything around us can be described in terms of particles and waves!
     We need to learn how to deal with waves.

 •   Examples of waves: sound, light, water waves, earthquakes, ‘Mexican’
     wave, etc.

 •   Later you will learn about other type of waves, in which “fields” do the

 What is common in all these different waves? What is common in sound,
 light, water waves, earthquake, etc?

 Answer: The math! The same type of equations can describe all these waves.

     •    The math in all cases describes a disturbance of the given medium that
         propagates from one region to another.

     •    While the disturbance propagates, the material stays near its
         equilibrium position. What propagates with the wave is energy and

             •   Wave pulse: eg transverse wave-pulse set up in a taut rope

             •   It is possible to generate a train of wave pulses: eg hammering,
                 periodically throwing pebbles into water, etc

             •   In many cases, the train of wave pulses is extremely long, with
                 large number of repetitions. In fact, in most of our calculations
                 we shall assume that the wave train is infinitely long. We’ll say
                 it is perfectly periodic.

             •   Transverse and longitudinal waves.

 Unit 1 Waves                                                                 3

http://www.physics.nwu.edu/ugrad/vpl/waves/wavetypes.html (local copy)

http://members.aol.com/nicholashl/waves/movingwaves.html (local copy)

http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html (local copy)

Other assumptions:

            •   We’ll only deal with one-dimensional waves (waves that travel
                along only one axis)

            •   The function describing the wave depends on the coordinate
                (say x) and time (t): f(x,t) (Often the Greek letter R(x,t) is used
                to describe the wave).

            •   Waves that do not change their shape as they travel

What do we know about the function, f(x,t), that describes the wave?

       •    It is a periodic function of space:

                                 f(x,t) = f(x+8, t)

       where 8 is called the “wavelength”.


Unit 1 Waves                                                                  4
           Later on we shall often use the term “2B/8”. Let’s define a term:

                                           k = 2B/8

           for shorthand. We’ll call this the angular wave number. Its units
           are [k] = rad/m

       •   f(x,t) is also a periodic function of time:

                                     f(x,t) = f(x, t+T)

           where T is called the “period” of the wave; units of period: [T] =

       •   In the math that follows, we shall often use the term ‘1/T’. We
           define a new “shorthand’ term that we call the frequency, f:

                                       f = 1/T

           Units: [f] = 1/s = Hz (Hertz). (Sometimes, instead of ‘f’ we use the
           Greek letter ‘ <‘ for frequency.)

       •   We also define a term, which is analogous to frequency, but
           measures the angular change per time. We call this the angular

                                     T (omega) = 2Bf

           Units [T] = radians/sec

       •   In one period, the wave travels one wavelength, so the speed of the
           wave can be defined as:

                                      v = 8/T

           which is the same as

                               v = 8f = (8/ 2B) (2B f) = T /k

           Units of speed [v] =m/s.

       •   The wave speed … speed of particles in the wave!

What about the amplitude of a wave?

       •   It depends on the type of wave we are talking about.

       •   In sound waves it can be the pressure, in light waves it is the
           magnitude of the electric field, in some quantum phenomena it can

Unit 1 Waves                                                               5
           be the probability amplitude. So for each type of wave the
           amplitude of the wave is different.

What about f(x,t)? What kind of function can describe a wave?

       •   ‘Looks like’ sin(x,t) or cos(x,t). Does that make sense?

       •   Yes, cos or sin functions can describe waves.

       •   More precise answer: Is it always OK to use harmonic functions to
           describe waves because (according to the Fourier theorem) any
           periodic function can be written as the sum of harmonic functions:

                               f(x) = 'ansin(knx)

For example:

http://www.phy.ntnu.edu.tw/java/sound/sound.html (no local copy)

From what we have learned so far, we can write the equation of a wave as:

                            f(x,t) = Asin(ax+bt)

where ‘x’ is distance, ‘t’ is time, ‘A’ is the amplitude of the wave, and ‘a’ and
‘b’ are yet to be determined.

What do ‘a’ and ‘b’ represent?

       •   Assume that t = 0 (ie. we take a snapshot of the wave at t=0):

                                 f(x,0) = f(x+8,0)

                          Asin(ax,0) = Asin(ax+a8,0)

                                ax = ax + a8 "2B

                                    ˆ a = 2B/8

           Therefore a = k = 2B/8 angular wave number.

       •   Assume now that at x = 0:

                               f(0,t) = f(0,t+T)

Unit 1 Waves                                                                 6
                         Asin(0,bt) = Asin(0, bt+bT)

                                bt = bt + bT "2B

                                   ˆb = 2B/T

Therefore b = T = 2Bf angular frequency.

Now we are ready to specify the general equation of a wave:

                          f(x,t) = Asin(kx + Tt + N)

where A = amplitude, (kx + wt + M) is called the phase of the wave, and M is
the phase constant.

Other forms of the same wave:

                        f(x,t) = Asin(2Bx/8 +2Bf t+ N)

                         f(x,t) = Asin2B(x/8 +f t+ N)

                       f(x,t) = Asin2B/8(x +2Bv t+ N)

The next question we have to answer is: If the above equations describe a
wave that is moving along the x-axis, how do we know from these equations
which direction the wave is travelling?

The answer is simple: from the sign of the ‘Tt’ term. Let’s see why:

       •   We know that the actual shape of the wave does not change as the
           wave moves. That means that the phase of the wave is constant:

                              kx - Tt = constant

Unit 1 Waves                                                             7
         which means that the phase at a given time and given position is
         the same as the phase at a later time and position:

                                   kx’- Tt’ = kx - Tt

         Let’s denote x’ = x + )x and t’ = t+)t. If the wave is moving
         towards the positive x-direction, )x is positive. Thus for the phase
         to stay constant, the sign of the ‘Tt’ has to be negative. That is, a
         wave travelling toward larger x values (positive x) is describe by
         the function

                                 f(x,t) = Asin(kx - Tt)

         If we change the sign in front of the ‘Tt’ term, the function will
         describe a wave travelling in the negative x direction:

                                 f(x,t) = Asin(kx + Tt)

         (Note the ‘+’ sign in front of the Tt term.)

Unit 1 Waves                                                              8
Speed of waves in various media:

Waves on a string:

    Waves travel with different speeds in different media. Let’s calculate the
    speed of travelling waves on a tightly stretched string, and learn some
    general concepts from it.

          a. Assume that the wave moves with a constant speed v to the

          b. The string under constant tension Ftension

          c. The mass density of the string is : kg/m

          d. The amplitude of the wave is small

          e. We’ll consider the string in the wave frame of reference; that is
             we are travelling along with the wave. If the wave is moving to
             the right, from the wave frame of reference, the string will
             move from left to right:


                                      A        B

(textbook p:454)

Unit 1 Waves                                                             9
        •   We’ll look at a small segment of the string AB that makes an
            angle )2 .

               o We shall approximate this segment as an arc of circle of
                 radius R.

               o The length of the segment is R)2

               o The mass of this segment is given by the mass density, :,
                 times the length of the segment:

                                         Mab = R)2:

               o Looking at the string from the frame of reference of the
                 wave, the tiny segment of the string is essentially revolving
                 around point C with speed v

               o Since it is moving at a constant speed around a radius ‘R’,
                 the net force acting on this segment is equal to the
                 centripetal force acting on the string, which must come
                 from the y component of the tension in the string:

                              2(Ftension)y = Mabv2/R

               o For small angles, sin2 . 2, hence:

                       2Ftsin(½ )2) = Ft)2 = Mabv2/R =

                              (R)2:)v2/R = )2:v2

               o Rearranging this we get:

                                   V =


        •   Speed depends on the tension, Ft, in the string, and the mass density
            (mass per unit length) of the string.

        •   Speed does not depend on the frequency of the wave

How about the speed of waves in other media? Things like sound waves in air
or a fluid?

Unit 1 Waves                                                               10
Speed of compression waves

Compression waves: imagine a number of ping-pong balls attached by small
springs all placed on a table:

If we start the first ball on the left to vibrate in simple harmonic motion, the
other balls will follow suit. Propagation of compression waves occurs along
the direction in which the balls oscillate, and is characterised by a series of
alternating high and low density of balls. This models is OK for atoms in an
elastics substance and/or a liquid.

See for example:
http://www.physics.nwu.edu/ugrad/vpl/waves/wavetypes.html (local copy)

It can be shown (see textbook p:459) that the speed of sound for compression
waves in liquids is:

                                    V =

where B is the Bulk Modulus defined as

                                B = -)P/()V/V)

and D is the density of the liquid. This expression may look different from the
one we derived for a wave on a string, but it is not. If you look at the units B
(kg/m.s2) and that of D (kg/m3) , you find that it is the same as that for the
speed of a wave on a string, that is (force/density)1/2 . We’ll look at example,
when we talk about sound waves in a later unit.

Unit 1 Waves                                                                11
Reflection, transmission and absorption
When waves arrive at the interface between two media, they undergo
reflection, transmission and/or absorption. For example, when light waves
arrive from air to a window, part of the light wave will be reflected, part
will be transmitted, and a (small) part will be absorbed.

Let’s try to model these phenomena with mechanical waves, eg. a wave-pulse
travelling on a rope (pictures from textbook pp: 456-457):

Unit 1 Waves                                                             12
(see for example,

Unit 1 Waves                                                    13
Unit 1 Waves   14
Energy (power) carried by waves
Question: do waves carry energy? Light waves do (eg. solar cells), sound
waves do (our eardrums move), etc. How about waves running on a string?

Let’s shake a piece of string up and down. How much energy travels in the
string to the other end?

Let’s calculate it:

As we move our hands up and down at a point ‘x’ with force Fy in the y-
direction, we do work. The power involved is:

                                     P = Fy vy

where ‘vy’ is the speed of the string in the y-direction, and Fy is the force in the

                       Fy = Fsin2 = F2 (for small angles)

Since we know the function that describes the wave, we can calculate the
speed,vy, and the angle, 2, at any point on the string:

                               y(x,t) = Acos(kx-Tt)

                           vy = dy/dt = ATsin(kx-Tt)

                      2 . tan2 = dy/dx = -k Asin(kx-Tt)

So the power travelling in the string is

             P = Fy vy = F2 vy = F (k Asin(kx-Tt)) (ATsin(kx-Tt))

                              = A2kFTsin 2(kx-Tt)

Using v = T/k and F = :v2 we find that the power flowing past the point ‘x’ at
a given time, t, is:

                            P = A2:v T2 sin 2(kx-Tt)

This is the instantaneous power. We are often more interested in the average
power. So we calculate the power in one cycle (integrate over one cycle):

                               Paverage = ½ A2:v T2

Unit 1 Waves                                                                 15
This is an important result: the power carried by a wave is proportional to the
amplitude squared! This is true for other type of waves, not only waves on a
string. This is something we’ll discuss when we start with quantum physics.

Interference of waves

Question: What happens when two or more waves pass through the same
region at the same time? In the case of particles, they “see” each other and
scatter (eg. billiard balls). Do waves behave similarly? Do they change each
other’s path?

Answer: Waves do not interact with each other! When two or more waves
overlap, each wave “does it’s own thing”. The resultant wave is the algebraic
sum of the primary waves. This is called the Superposition Principle.

(textbook p: 465)

Unit 1 Waves                                                             16
Let’s look at specific examples:

a. Two waves: y1 and y2 have identical amplitude, frequency and wavelength
   but different phase constant:

                   y1 = Acos(kx+Tt) and y2 = Acos(kx+Tt +N)

The new wave will be the algebraic sum of these two:

                yT = y1 + y2 = Acos(kx+Tt) + Acos(kx+Tt +N)


                   cos" + cos $ = 2cos [("+$)/2] cos[("-$)/2]

we get:

            yT = 2Acos(kx+Tt+N/2)cos(N/2) = A*cos(kx+Tt+N/2)


                               A* = 2A cos(N/2)

What this means: The new wave, yT, has the same frequency and wavelength
as the original waves, but the amplitude of the new wave, A*, depends on the
phase constant difference of the original two waves, N .

For example:

                              if N = 0   A* = 2A

                               if N = B A* = 0

In general when 0 # N # B , the new amplitude is

                                   2A # A* # 0

We call this phenomenon wave interference, or just interference. When

                     A* < A => destructive interference


                     A* > A => constructive interference

See, for example
http://www.gmi.edu/~drussell/Demos/superposition/superposition.html (local copy)

Unit 1 Waves                                                              17
(textbook :464)

(b) Now let’s add two waves that have the same amplitude, phase constant but
slightly different frequency:

                  yT = y1 + y2 = Acos(kx + T1t) + Acos(kx + T2t)


                                  T2= T1 + )T

and )Tis very small.

                  yT = Acos(kx + T1t) + Acos(kx + (T1 + )T) t)

To make things simpler, let’s assume x = 0:

                        yT = A[cos(T1t) + cos(T1 + )T) t]

Using (again)

Unit 1 Waves                                                          18
                   cos" + cos $ = 2cos [("+$)/2] cos [("-$)/2] ,

we find:

                       yT = 2Acos[(T1 + T2)t/2]cos()Tt/2)

which can be written as

                                  yT = A*cos(ft)

where f is the average frequency:

                                  f = (T1 + T2)/2

What this means is that by adding two waves of slightly different frequencies,
we get:

    (i)     a new wave whose frequency is the average frequency of the
            starting two waves

    (ii)    the new amplitude, A*, that varies in time

We call this the beating of two waves. Easy to observe for sound, but it is also
true for all type of waves. The beat frequency is defined as:

                                    fbeat = f1 -f 2

(see for example, http://webphysics.ph.msstate.edu/jc/library/15-11/index.html (local
copy) or http://www.gmi.edu/~drussell/Demos/superposition/superposition.html
(local copy))

(textbook p:465)

(c) Two identical waves travelling in the opposite direction:

Unit 1 Waves                                                                   19
                               y1 = Asin(kx - Tt)

                               y2 = Asin(kx + Tt)

The sum of the two waves results in

                  yT = y1 + y2 = A[sin(kx-Tt) + sin(kx+Tt)]=

                            yT = 2A sin(kx)cos(Tt)

Please note that since wave phase is not (kx " Tt) type, this is not a travelling
wave. If we plot this wave:

This is not a travelling wave, it is a standing wave.

Unit 1 Waves                                                                20
Standing waves

           •   Nodes: positions on a standing wave where the amplitude is
               always 0. Nodes occur when

                              y(x) = Asin(kx) = 0

                         ˆ kx=nB      where n =0,1,2, ...

               That is

                                   2Bx/8 = nB

               which means nodes occur at positions:

                                   xn = n8/2

           •   Anti-Nodes: positions where the amplitude is maximum. Anti-
               nodes occur when

                                   sin(kx) = 1

                                 ˆ kx=(n+1/2)B

                               2Bx/8 = (n+1/2)B

               That is, anti-nodes occur when:

                         xn = (n+1/2)8/2 = (2n +1) 8/4

http://www.gmi.edu/~drussell/Demos/superposition/superposition.html (local copy)

Unit 1 Waves                                                               21
Assume a string with both ends fixed. If we generate waves on this string,
waves will bounce back and forth, thus very soon we’ll get standing waves
emerging. Since both ends are fixed, the end points are nodes. Only certain
waves satisfy this ‘boundary condition’. The wavelengths of the waves that
can exist on a string with both ends tied down are:

                                    8n = 2L/n

where n = 1,2,3,...and L is the length of the string

(textbook p:480)

Unit 1 Waves                                                           22
    The frequencies is these waves are:

                                    fn = v/8n = nv/2L

    where n is an integer.

    Since we know how to calculate the speed of a wave on a string, the
    frequencies can be also written as:

                                            n     Ftension
                                    fn =
                                           2L        µ

    The lowest frequency (longest wavelength) is called the fundamental mode or
    first harmonic. The next allowed frequency is the second harmonic, followed
    by the 3rd harmonic, etc


               •   Standing waves with boundary conditions result in discrete
                   frequencies! (This is true for all type of waves, not only waves
                   on a string)

               •   Discrete frequencies => frequencies ‘quantized’

    Question: What happens if we try to generate all type of waves on a string with
    2 ends fixed?

    Answer: We find that only the ‘allowed’ frequencies will be excited. The
    amplitude of the other vibrations will stay very low:


                               fn          fn+1       fn+2     frequency

    These ‘special’ frequencies are called resonant frequencies.

    Unit 1 Waves                                                              23

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