Module 2: Waves

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By: CoryMathews 123
February 10, 2009 | Physics - Mirley Balasubramanya
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Outline

  1. What is a wave?
  2. Characteristics of a Wave
  3. Classifiication of Waves
  4. Wave Speed
  5. Principle of Superposition
  6. Standing waves on a string
  7. sound as a wave
    1. Speed of Sound
    2. Standing Waves - Musical Instruments
    3. Loudness of Sound
  8. Doppler Effect
  9. Shock waves
  10. Applications of Waves

1. What is a wave?

Oscillations, Up/Down or Back/Forth, Motion, Patterns, Period, Ocean(Medium)
A pattern that propagates.
No matter moves with the wave. Only the pattern moves.


2. Characteristics of Waves

Most waves are made up of cycles. Called a wave train (a train of cycles).

Cycle length = wave length
Symbolized by lambda(λ)

Period = Time of 1 Cycle or 1 wave length
In response to the wave particles of the medium oscillate periodically. Period is also the time for one oscillation.
Frequency  is the # of periods in 1 second. Measured in Hz.

Relation between V (Wave speed), λ and f
V = fλ

Though this formula contains f and λ it does not depend on them.

 


 

3. Classification of Waves

Transverse Wave
A strait string is wiggled up and down at one end sending a wave down the string. The direction of the wave is down the string while the direction of each particle is perpendicular to the direction. This is called a transverse wave.

Longitudinal Wave
A slinky is wiggled left and right at one end sending a wave down the string. The direction is the same as the direction of the particles. This is called a longitudinal wave.

Another Classification
String wave is considered a 1-Dimensional wave. It propagates along a strait line.
Water waves are 2-dimensional waves. It "ripples" in all directions on a plane.
Sound is a 3-Dimensional wave. It travels in all directions.

Mechanical and Electromagnetic Waves
Mechanical Waves Requires a medium. If you tried to talk in a vacum noone would be able to hear because there is no air to carry the sound.
Electromagnetic Waves do not need a medium.

 


 

4. Wave Speed

V is determined by the characteristics of the medium.

Examples:
A string is attached to a wall at 1 end and a weight hanging over a pulley on the other end. A wave is then created.
V depends on tension, linear mass density (μ) = Mass per unit length

Sound is effected by Temperature.

 

Example:
(A) The density of iron is 8800kg/km3. An iron wire 2m long is under a tension of 1000N. Its thickness is 2mm. What is the transverse wave speed in this string?
(B) If the frequency is 10Hz, what is its wavelength?
(A)
We have a wire length L,  with a diameter d.
we know that d = 2mm and from this the radius r = d/2 = 1mm.
m = mass of wire of length L.
volume, V = (pi*r2)L
Density = m/v = m/(pi*r2)L
The linear density μ = m/L
From the density we see that Density = μ / pi*r*2
μ = (Density)pi*r2
μ = (8800)*pi*(.0001m)2
v = sqrt((Ft = 1000)/μ) = 190.2m/s
(B)
f = v/λ
λ = v/f = 190.2m/s / 10Hz = 19.02


5. Principle of Superposition

If a pulse (pulse 1) is sent down a string from left to right and another pulse (pulse 2) is send from right to left, they will simply pass right by each other and at the point where they meet the pulse will be both of them added together. They do not destroy each other or lower they velocity.

Principle: Let y, be the displacement produced by wave 1 at a point and the displacement produced at the same point at the same time, each acting independently. The net displacement is y1 + y2 when they act together. A non physics example would be if a couple is not married and each donated 10,000 to charity. Then they get married and donate 20,000 to charity.

When both pulses are aiding each other (ex. both push up) then they are called Constructive Interference.
When one pulse pushes up and the other down at the same powers it is called Destructive Interference. 


 6. Standing Waves on a String

Standing waves are produced when two waves of the same length travel in the same medium.

Patterns in standing waves.

 Given a string length L fixed at both ends. You then hit the string giving it a Node(N) at both ends and an Anti-Node(A) in the middle.
There is a single segments in Length L. 
This is the first harmonic standing wave (Or fundamental).
When the string has two nodes at the ends and then one also in the middle then it is considered the second harmonic. (This was done in the lab)

Example:
A string 1.5m long has an oscillator attached to one end and 100 grams hanging from the other end.
a) What are the standing wave wavelengths?
b) What are the standing wave frequencies?
c) What is the travelling wave Speed?
a)
3.0, 1.0, 1..
b)
Ft = .1*9.8 = .98N
c)
V = sqrt(Ft/μ) = sqrt(.98/.004) = 15.65
f1 = v/λ1 = 15.65/3 = 5.22Hz
f2= v/λ2= 2(5.22)= 10.44Hz
f3= v/λ3= 3(5.22) = 15.66Hz

Stringed Musical Instruments
Producing different tones
1. Change Tension = change in velocity (V)
    λ is fixed because L is fixed.
    frequency changes because f = v/λ



 

7. Sound as a Wave

Sound is a mechanical longitudinal wave. Mechanical meaning it needs a medium (air).

 

7.1 Speed of Sound

A) Solids

Density of a solid (P)
Y is the Youngs Modulos.

vSolid = sqrt(Y/P)

Ysteel = 200x109Pa
Psteel = 7850kg/m3
Vsteel = sqrt(200x109/7850) = 5047m/s
Sound can go through steel at 5047meters/second or 3.14 miles a second

B) Liquid

Force/Area = Pressure (P)
P = (Constant)(Δv/v)
The constant is a bulk modulus of liquid (Elastic Property)
The Bulk Modulus of water is 2.2x109Pa

Vliquid = sqrt(B/P)

Vwater = sqrt(2.2*109/1000kg/m3) = 1483m/s 
Sound can go there water at about 1483m/s or .92 miles per second, slower then steel.

C) Gas

Ideal Gas Law: PV = nRT or Pressure * Volume = (# of moles of gas)(Gas Constant)(Kelvin Degrees)
Gamma = γ = (specific heat at Constant Pressure)/(Specific heat at constant volume)
Vair = (331.5m/s)sqrt(1+((Temp in Celcius)/(273.15)))

Vsolid > Vliquid > vGas

 

7.3 Loudness of Sound

 Physical Entity -> Sensation in Brain
Physical Entity can be measured while Sensation in Brain is assessed (Louder to one person then the other)

Sound Frequency -> Pitch
Light Frequency -> Color
Intensity of Sound -> Loudness

Intensity of Sound 
Power = Energy/time
Watt = Joule/second

Wave Power = Energy Carried through a cross section in unit time.

Intensity of Wave (I) = Power(P)/Area

What is the least intense wave heard by the human ear? 10-12W/m2

Io = 1 x 10^-12
dB = 10*log10(I/Io

Distance dependance of Intensity 
Given a speaker with Power P find the Intesity at a distance of r (Radius).
I = P/4*pi*r2
Intesity is inversely propertional to the square of the distance.

Decibel Level or Intesity Level "measures" or assesses loudness in deciBells(dB)
Intesity Level or Decibel Level = β = 10log(I/10-12)

Intensity to Loudness in  dB
10-12 = 0dB
10-11 = 10dB
10-10 = 20dB

Exposure to 85dB or greater will damage your hearing over time.

Example: 
A typewritter produces 60dB of sound.
You have 10 typists what is the total decibell level in the room?
β = 10I
βtotal = 10log(10I/Io)
βtotal = 10[log10 + log(10/Io)
= 10 * β
βtotal = 10dB + 60dB = 70dB


 

 

 

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