One formula for a wave with a y displacement (e.g., of astring) traveling in the x direction is
y(x,t) = A\sin(kx-\omega t).

All the questions in this problem refer to this formula and tothe wave it describes.
a) Which are independent variables?
b) Which are parameters that determine the characteristics ofthe wave?
c) What is the phase phi(x,t) of the wave?
d) What is the wavelength lambda of the wave?
e) What is the period Tof this wave?
f) What is the speed of propagation v of this wave?

The independent variable is the variable whose value change (controlled or not controlled) and that does not depend on the other variable
The dependent variable is the variable whose value depends on the independent variable

For the wave in this problem, therefore, we have:

The time (t) and the x-position of the wave (x) are the two independent variables
The displacement along the y-direction (y) is the dependent variable, since its value depends on the value of x and t

b)

In physics and mathematics, the parameters of a function are the quantities whose value is constant (so, they do not change), and the value of the dependent variable also depends on the values of these parameters.

Therefore in this problem, for the function that represents the y-displacement of the wave, the parameters are all the constant factors in the formula that are not variables. Therefore, they are:

k, called the wave number
A, the amplitude of the wave
[tex]\omega[/tex], the angular frequency of the wave

c)

The phase of this wave is zero.

In fact, a general equation for a wave is in the form

[tex]y(x,t)=Asin(kx-\omega t+\phi)[/tex]

where [tex]\phi[/tex] is the phase of the wave, and it represents the initial angular displacement of the wave when x = 0 and t = 0.

However, the equation of the wave in this problem is

[tex]y(x,t)=Asin(kx-\omega t)[/tex]

Therefore, we see that its phase is zero:

[tex]\phi=0[/tex]

d)

The wavelength of a wave is related to the wave number by the following equation

[tex]k=\frac{2\pi}{\lambda}[/tex]

where

k is the wave number

[tex]\lambda[/tex] is the wavelength

For the wave in this problem, we know its wave number, [tex]k[/tex], therefore we can find its wavelength by re-arranging the equation above:

[tex]\lambda=\frac{2\pi}{k}[/tex]

e)

The period of a wave is related to its angular frequency by the following equation

[tex]\omega=\frac{2\pi}{T}[/tex]

where

[tex]\omega[/tex] is the angular frequency

T is the period of the wave

For the wave in this problem, we know its angular frequency [tex]\omega[/tex], therefore we can find its period by re-arranging the equation above:

[tex]T=\frac{2\pi}{\omega}[/tex]

f)

The speed of propagation of a wave is given by the so-called wave equation:

[tex]v=f\lambda[/tex]

where

v is the speed of propagation of the wave

f is the frequency

[tex]\lambda[/tex] is the wavelength

The frequency is related to the period of the wave by

[tex]f=\frac{1}{T}[/tex]

So, we can rewrite the wave equation as

[tex]v=\frac{\lambda}{T}[/tex]

From part d) and e), we found an expression for both the wavelength and the period:

[tex]\lambda=\frac{2\pi}{k}\\T=\frac{2\pi}{\omega}[/tex]

Therefore, we can rewrite the speed of the wave as:

[tex]v=\frac{2\pi/k}{2\pi/\omega}=\frac{\omega}{k}[/tex]

Learn more about waves:

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