assuming that the wave’s frequency ω remains constant, create a plot of the wavelength λ as a function of the current speed u. as u increases, what do you expect to happen to i) the wave’s length? ii) the wave’s amplitude? iii) the wave’s steepness? (5p)

Wavelength(λ) will increase with increasing velocity. The graph is attached below, where x-axis indicates wavelength and y-axis indicates velocity.
Wave’s amplitude will remain unaffected with increasing velocity.
The wave’s steepness will decrease with increasing velocity.

What are wave characteristics?

A wave is a propagating dynamic disturbance of one or more quantities. Waves can be periodic, in which these quantities repeatedly oscillate around their equilibrium values at specific frequencies.

Whether vibrations or waves, they can all be characterized by four characteristics: Amplitude, Wavelength, Frequency, Velocity.

u = ωλ

Where, u = velocity of the wave

ω = frequency of wave

λ = wavelength of wave

For the first case:

If the wave’s frequency ω remains constant, then the velocity of wave will be proportional to wavelength:

u ∝ λ

So, if the velocity increases then there will be an increase in wavelength.

For the second case:

The amplitude of the wave is completely independent of the speed of propagation u and depends only on the amount of energy in the wave.

So, there will be no change of amplitude.

For the third case:

Wave steepness is the ratio of wave height(H) to wave length(λ).

Wave steepness = H/λ

This indicates that wave steepness is inversely proportional to wavelength.

Wave steepness ∝ 1/λ

But we know,

u ∝ λ

Then,

Wave steepness ∝ 1/λ ∝ 1/u

So, according to above application wave steepness in inversely proportional to velocity. This means that steepness will decrease with increasing velocity.

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