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23 Nov 2019

Observe surface waves on water. Do they seem to be dispersionless? Do they obey the superposition principle? Does the degree to which they obey the superposition principle depend on the amplitude of the waves?

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Consider a traveling wave described by the formula

$y_1(x,t) = A \sin(k x - \omega t)$ .

This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

Part A

Which one of the following statements about the wave described in the problem introduction is correct?

ANSWER:

The wave is traveling in the

direction.
The wave is traveling in the

direction.
The wave is oscillating but not traveling.
The wave is traveling but not oscillating.

Part B

Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time t=0

this new wave should have the same displacement as y_1(x,t)

, the wave described in the problem introduction.

ANSWER:

$A \cos (k x - \omega t)$
$A \cos (k x + \omega t)$
$A \sin (k x - \omega t)$
$A \sin (k x + \omega t)$

The principle of superposition states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves , where is the wave described in Part A and is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

$y_{\rm s}(x,t) = y_{\rm e}(x) y_{\rm t}(t)$ .

This form is significant because , called the envelope, depends only on position, and depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of .

Part C
Find and . Keep in mind that should be a trigonometric function of unit amplitude.

erinhyena808

Which statement about seismic waves is TRUE?

A) Only S waves can travel through liquid.

B) P waves travel faster, and thus arrive sooner than do the S waves.

C) In the shadow zones, neither type is observed.

D) S waves can travel through the outer core, but P waves cannot.

E) On the far side of the Earth, only the S waves on the surface can be detected.

lavenderdonkey844

A fisherman notices that his boat is moving up and downperiodically, owing to waves on the surface of the water. It takes2.70 s for the boat to travel from its highest point to its lowest,a total distance of 0.600 {\rm m}. The fisherman sees that the wavecrests are spaced 8.00 m apart.
A.How fast are the waves traveling?
B.What is the amplitude of each wave?
C.If the total vertical distance traveled by the boat were 0.500 m,but the other data remained the same, how fast are the wavestraveling ?
D.If the total vertical distance traveled by the boat were 0.500 m,but the other data remained the same, what is the amplitude of eachwave?

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Observe surface waves on water. Do they seem to be dispersionless? Do they obey the superposition principle? Does the degree to which they obey the superposition principle depend on the amplitude of the waves?

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Physics: Principles with Applications

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