Lecture 2: Wave Phenomena II and Adding up waves.

Sound is a type of wave. By hitting the keys on a piano we make the strings vibrate, this will make the air around it change its pressure, which is what we hear as sound. We will use what we just learned last class to understand how we can make a piano string vibrate at the frequency we want. This will produce the various pitches we hear.

The parameters of the harmonic wave function.

Recall, the displacement y depends on x,t (it is a 2-D function)

We represent it here by 2 1-D functions. Space dependence Time dependence

0

2 2sin

2 20 2 / 2

4 3sin

y y A x tT

y x t

π π φλ

π π π

⎛ ⎞= + ± +⎜ ⎟⎝ ⎠

⎛ ⎞= + − +⎜ ⎟⎝ ⎠

Wave velocity

Wave velocity: formula

The peak moved a distance λ in the time T, so v=d/t = λ/T

Conceptual Question:Displacement vs position in a Longitudinal Pulse wave

A wave is sent along a long spring by moving the left end rapidly to the right and holding it fixed.

Which graph correctly shows the relation between displacement as a function of position at the instant shown, t2?

Measure displacement relative to the slinky’s position at t1.

Assume displacement to the right is positive.

t2

t1

Sound Waves

A vibrating object acts as the source of the wave. As a speaker moves through a cycle, it fixes the period and frequency.Speed of sound ~ 343 m/s

The frequencies of a piano.

Lowest f = 27.5 Hz Highest f = 4186.0 Hz

Octaves: when you change by twice the frequency, or a ratio of 1:2

Twelfth 1:3 Fifth 2:3

Producing the desired frequency

Let’s apply what we learned: v = λf, in general. The speed depends on medium

properties. v= √(F/), for a wave on a string or rope.

F is tension, is mass density (m/L). Therefore, we can solve for the frequency:

λf = √(F/)

The denser the string, the lower the frequency The longer the string, the lower the frequency Adjusting the tension can help us do small

adjustments, or fine tuning.

1 Ff

λ =

Allow a longer wavelength.

The longer the wavelength, the lower the frequency.

1 Ff

λ=

Tension

Adjusting the tension can help us do small adjustments, or fine tuning.

1f

F

λ =

Mass Density

Varying the mass density can change the frequency. By going to a denser, thicker cable we can get a lower frequency: Bass string.

1 Ff

λ=

What would happen if we couldn’t change ?

Use a wire that’s about 2 in long to give me the highest frequency of a piano (4186 Hz).

Keep the tension about the same and use the same wire.

How long should the wire that will give us the lowest note (27.5 Hz) be?

Adding waves

What happens when we have several disturbances in the medium? Wave 1 creates displacement y1

Wave 2 creates displacement y2

Total displacement will be yTOT=y1+y2

Principle of Superposition Will be valid as long as Hooke’s Law

is valid (F=-kx)

Q: will the resulting amplitudes always be greater?

Conceptual question:Wave addition

True or False: Since the displacements add when

superposing two waves, the amplitude resulting of the superposition of two waves will always be greater than the amplitude of either of the original waves.

Superposition of oppositely traveling wave pulses.

ConstructiveInterference

DestructiveInterference

Standing waves: adding waves travelling in opposite directions.

The picture above shows a “standing wave”. We will study how we can produce such waves by adding one wave to another. Piano strings, guitar strings, bass strings, all of these make sound using standing waves.

Superposition of 2 traveling harmonic waves, as a function of time.

The period and wavelength are exactly the same.

One wave travels to the right, one to the left.

Standing Wave

Plucking the string in the middle, it will vibrate.

Note: wavelength in the picture is twice the string length: λ=2L

Fundamental mode

2nd Harmonic

3d Harmonic

Standing waves: one loose end

Standing waves and harmonics

The patterns you just learned are the basis for musical sounds.

These same patterns occur are the basis for the structure of atoms, the periodic table and therefore all of chemistry! To be discussed quantum

mechanics…in a few weeks