Waves

A fisherman is fishing from a boat at rest. A speedboat passes close by. The wake of the boat passes across our fisherman. What happens? The wave disturbs our fisherman. He experiences the energy of the wave as it passes through him, but not the bulk of the water.

Wave - traveling disturbance that transmits energy without transferring matter

Types of waves:

Electromagnetic
- Examples -- light waves, radio waves, microwaves, X-rays, etc.
- Do not require a medium for transfer; can be transferred through a vacuum
Mechanical
- Examples -- sound waves, water waves, etc.
- Require medium for transfer; cannot be transferred through a vacuum
- The speed of the wave depends upon the mechanical properties of the medium.
- Some waves are periodic (particles undergo back and forth displacement as in a sound wave.)
- Some waves are sinusoidal (paricles undergo up and down displacement as in a wave on a string.)

Types of mechanical waves:

Transverse
As a wave passes through a point, the particles vibrate at right angles to the direction in which the wave is moving
Notice -- as the wave moves through point A, the particle does not return to its original position until point E
Crest
upward displacement of transverse wave
Trough
downward displacement of transverse wave

Speed of a transverse waveThe speed of a transverse wave depends upon the tension or force the medium is subjected to and the mass per unit length of the medium (mass/length). If the tension or force is increased, the speed of the wave increases; if the mass/length of the medium increases, the speed decreases.

Longitudinal
As wave passes through a point, the particles vibrate parallel to the direction in which the wave is moving

Speed of a longitudinal waveThe speed of a longitudinal wave depends upon the properties of the medium. If the density of the medium is increased, the speed of the wave decreases; if the modulus E (a constant that is characteristic of the medium that indicates how the medium reacts to stress) of the medium increases, the speed increases.

Speed of a longitudinal wave formula

An applet that can help you visualize the difference between a transverse wave and a mechanical wave.

Another good wave applet

Wave terms:

Wavelength
the distance between two successive in-phase points; symbol is l and SI unit is meters
Amplitude
maximum displacement of wave; measure of wave's energy
Pulse
single disturbance of a medium
Frequency
the number of waves passing a point per second; symbol is n or f and SI unit is Hertz (Hz)
Period
time for one wave; symbol is T and SI unit is second
T = 1 / n or T = 1/f
n = 1 / T or f = 1/T
Speed
the speed with which the wave moves through the medium is the product of the wavelength and the frequency; SI unit is m/s
v = l n or v = l f

Wave properties

Applet showing reflection, refraction, & diffraction

Reflection
"bouncing back" of a wave
- Laws of Reflection - The angle of incidence equals the angle of reflection (or it can be stated q₁ = q₂) ; the incident wave, the reflected wave, and the normal all lie in the same plane
- Angle of incidence and angle of reflection are always drawn relative to the normal
  Normal - line drawn perpendicular to surface
  Example of reflection - echo
Refraction - "bending of waves"
- A wave passing from one medium to another medium of different density changes its speed causing it to bend
- The speed of the wave is greatest in the less dense medium
  $picture of refracted wave$
  Refraction applet
Interference - result of the superposition of two or more waves
- Superposition Principle - when two waves are in the same place at the same time, the displacement caused by the waves is the algebraic sum of the two waves
  An applet that helps you visualize how two waves in the same place at the same time superimpose.
- Constructive interference
  waves interfere, "adding" to produce a larger wave
  
  Destructive interference
  waves interfere, "adding" to produce a smaller wave
  
  Constructive and Destructive Wave Interference
  A cool interference applet
  Standing wave
  two waves with the same wavelength, the same frequency, and the same amplitude that are traveling through a medium in opposite directions interfere producing a standing wave
  Node
  point of zero displacement on a standing wave
  Antinode
  point of maximum displacement on a standing wave
  Interactive applet that allows you to create constructive and desctructive interference between two waves.
  
  Tacoma Narrows Bridge Collapse
  Beats
Diffraction
spreading of waves around a barrier
Diffraction Applet

Additional Waves Applets

Waves at boundaries between different media:

speed of wave doesn't depend upon frequency or amplitude of wave; speed depends upon the properties of the medium
at a boundary, part of the incident wave is reflected back upon itself in the original medium and part is transmitted through the second medium
the more dense the second medium is, the less wave energy is transmitted
the reflected wave, the incident wave, and the transmitted wave all have the same frequency; the velocity and the wavelength of the waves change
when a wave passes from a less dense to a more dense medium, the reflected wave is inverted
when a wave passes from a more dense to a less medium, the reflected wave is unchanged

Excellent applet that allows you to visualize how a single pulse reflects at a fixed end and at an open end.

Mathematical Description of a Wave

A wave function describes the position of any particle in the medium at any time. We will consider a sinusoidal wave. The displacement of a particle is given by

y = A sin wt

where w is the angular frequency and A is the amplitude

Angular frequency

w = 2 pn or w = 2 pf

The wave disturbance travels from x = 0 to some point x to the right of the origin in a time given by x/v (from d = vt), where v is the wave speed.

y(x,t) = A sin w [ t - [x/v}]

Wave numberWe can define a quantity called the wave number or propogation constant, k (k is the number of waves per meter; if the wave is traveling in the -x direction, k would be positive.)

k = (2 p/l }

Since v = l n , then w = v k.

The wave function can be written as

y(x,t) = A sin (w t - k x)

It is important to remember that when using a calculator to evaluate the wave function, the calculator must be set to its radian mode.

Simple Harmonic Motion

When a vibration or oscillation repeats itself, back and forth, over the same path, the motion is described as periodic (example would be a spring). In a spring, the mass attached to the spring oscillates back and forth about the equilibrium position. When the mass stretches or compresses the spring, the spring exerts a force on the mass that acts to "restore" the spring to its equilibrium position. Hooke's law describes this restoring force, or F = -kx. Any vibrating system for which the restoring force is directly proportional to the negative value of the displacement x (or one that obeys Hooke's law) is said to be in Simple Harmonic Motion (SHM).

Terms which describe SHM:

DisplacementThe distance x of the object from the equilibrium position

Amplitude The maximum displacement from the equilibrium position.

Period (T) The time it takes for one cycle (or back and forth motion).

Frequency (f) The number of cycles completed per second (SI unit is Hz, or Hertz).

Energy in the Simple Harmonic Oscillator:

Work is done when a spring is stretched or compressed. The elastic potential energy of the spring is given by

PE = 1/2 kx²
Thus the total mechanical energy (E) is the sum of the kinetic and potential energies at that point.

E = 1/2 kx² + 1/2 mv²
The total mechanical energy (E) is equal to the elastic potential energy at the maximum amplitude of oscillation.

E = 1/2 kx² or E = 1/2 kA² where A is the amplitude
The maximum velocity (v_o) occurs at the equilibrium point. At this point, the maximum kinetic energy equals the total mechanical energy (E) of the oscillator.

E = 1/2 mv_o²
An object in SHM has its greatest acceleration at its greatest amplitude and its maximum velocity at its equilibrium point. The acceleration is given by

a = kx/m

Period of SHM The period of an object in SHM is dependent upon the stiffness of the spring (related to k) and the mass (m) that is oscillating. The period does not depend upon the amplitude.

A simple harmonic oscillator obeys Hooke's Law, F=kx. If a mass is hung on a spring and the spring is allowed to come to rest, the spring constant can be calculated knowing the mass and the displacement, x. If the mass is then pulled down and allowed to oscillate back and forth about the equilbrium point, the period of oscillation can be found using the above formula.

How waves occur on the AP exam:

Simple calculations using v=lf on multiple choice questions.
Multiple choice questions that require a basic understanding of the definitions of wave terms such as period, frequency, wavelength, beats, amplitude, etc.
Predicting superposition patterns of two wave pulses.
Predicting the properties of a wave reflected at a more dense boundary.
Free response questions that require calculations using v=l f.

Waves Sample Problems

Waves Homework

Waves & Sound AP Objectives