Chapter 15 One Dimensional Wave Motion

*In 1860 James Clerk Maxwell …ended up with a ______________________________ .

*In 1925 Louis De Broglie … proposed that electrons have a _______________________ .

page 15-3 Wave Pulses

A pulse is a single, localized disturbance that travels in a medium. A periodic wave is a repeated propagation of a disturbance through a medium (except for light) without any net displacement of the medium.

What type of wave can pass through a liquid or gas? transverse or longitudinal

Why?

What type of earthquake wave can pass through the core of the earth? transverse or longitudinal

page 15-4 Work exercise 1

Refer to Figure 2

*When an earthquake occurs, which moves up and down:

a) the mass with the pen attached b) the drum

Speed of a Wave Pulse (on a rope)

We will skip the derivation on pages 15-4 and 15-5. We will use the result

V = square root of (Tension/ m ) m is the mass per unit length

Why does the wave velocity increase when the tension increases?

Why does the wave velocity decrease when the mass per unit length increases?

pages 15-6 & 7 Dimensional Analysis We will skip this material but read it.

page 15-8 Speed of Sound Waves

We will skip the details and the equations.

Examine the table on page 15-9.

Why does the speed of sound in air increase as the air temperature increases?

Why is the speed of sound in solids greater than the speed of sound in gases?

page 15-10 Linear and Nonlinear wave motion

We will not try to analyze the turbulence present in nonlinear waves.

page 15-11 The Principle of Superposition

*The principle of superposition: If two waves occupy the same space at the same time then the net displacement of the medium is the sum of the individual wave displacements. (This assumes that the medium does not exceed its elastic limit.)

If both waves have the same sign of displacement then the net displacement will be larger than each individual displacement. *This situation is called constructive interference. The two waves are said to be in phase. If the displacements have opposite signs then the net displacement will be smaller than the magnitude of the largest displacement. *This situation is called destructive interference. The two waves are said to be out of phase.

Frame c) in Figure 6 represents ______________________________ .

How does wave motion differ from particle motion when waves "collide?"

page 15-12 Sinusoidal Waves

We will take the displacement of linear waves to be sine waves.

page 15-13 Wavelength, Period, Frequency

The wavelength, λ, is the distance between successive similar points on the wave

frequency, f Stand still and count the number of peaks of the traveling wave that cross your position every second. You have measured the frequency.

SPEED OF A WAVE

*The fundamental equation of wave propagation is V = l f . It is true for all waves.

page 15-14 Angular Frequency ω

ω = (2 π) / T

pages 15-14 & 15Spacial Frequency k note: this is not the force constant of a spring

We will use k in the next section.

page 15-16 Traveling Wave Formula

In Chapter 14 we discussed the calculation of the displacement as a function of time.

Y = A sin(ωt + φ) This expression shows the displacement that is seen by an observer at a fixed position.

1) Y = A sin(kX - ωt) and 2) Y = A sin(kX + ωt) both describe waves that are traveling along the X axis. These formulas allows you to examine the displacement at any X position at any time. Unfortunately, physics needs many letters to represent quantities. The "k" in this expression is NOT the force constant of a spring. It is the spacial frequency. k = 2π/λ

What are the units of kX? What are the units of ωt?

Examine equation 1 above. Let X be a fixed number. For a particular wave K and ω will be constants. Imagine that the time number is getting larger (i.e. we are considering future times). Describe the future values of the argument of the sin function (i.e. the angle).

Describe the future values of Y.

What direction is the wave traveling for equation 1?

What direction is the wave traveling for equation 2?

The velocity of the wave is V = ω/k .

page 15-17 Phase and Amplitude

The more general equation for the traveling wave is Y = A sin(kX – ωt + φ) .

We have already discussed amplitude and phase angle.

page 15-18 Standing Waves

DEMO We will observe a standing wave on a slinky.

*Why are the waves shown in figure 13 called "standing waves?"

*In Figure 13 you can see certain spots along the pattern that are brighter white than other points. What are these points called? Why are they bright in the photograph?

What creates a standing wave situation?

In Figure 14, what is being shown on the c) part of the diagram?

In Figure 14, describe the motion of the second and third dot positions in part c).

page 15-20 We will skip column a).

page 15-20 Waves on a Guitar String

DEMO Guitar

*What is the distance between two successive nodes (measured in units of the wavelength)?

(Is it 2* λ , one λ, ½ λ, 1/3 λ, ¼ λ etc. ?)

The standing wave with the largest wavelength is called the "fundamental."

Do you agree with the equation λ_n = 2L/n ?

page 15-21 Frequency of Guitar String Waves

Understanding the wavelength for the guitar string is key to understanding the frequencies, the tones, generated by a guitar string.

DEMO You will hear the separate harmonics for a single guitar string.

Each harmonic has its own frequency.

Recall: V = l f Also recall: V = square root of (Tension/ m ) m is the mass per unit length

Substitute for V: square root of (Tension/ m ) = l f

or f = square root of (Tension/ m ) / l

Are all of the l ’s the same for each harmonic?

How are the frequencies of the harmonics related to the fundamental frequency?

f_n = n*f₁

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