Answer:
a) The amplitud of the plain wave is 25, b) The wave length of the plain wave is [tex]\frac{\pi}{2}[/tex], c) The frequency of the plain wave is [tex]\frac{\pi}{60}[/tex], d) The speed of propagation of the plain wave is 0.082.
Step-by-step explanation:
A plain wave is modelled after the following mathematical model as a function of time and horizontal position. That is:
[tex]y(t, x) = A \cdot \sin (\omega\cdot t - \frac{2\pi}{\lambda}\cdot x)[/tex]
Where:
[tex]y(t, x)[/tex] - Vertical position wave with respect to position of equilibrium, dimensionless.
[tex]A[/tex] - Amplitude, dimensionless.
[tex]\omega[/tex] - Angular frequency, dimensionless.
[tex]\lambda[/tex] - Wave length, dimensionless.
After comparing this expression with the equation described on statement, the following information is obtained:
[tex]A = 25[/tex], [tex]\omega = 120[/tex] and [tex]\frac{2\pi}{\lambda} = 4[/tex]
a) The amplitude of the plain wave is 25.
b) The wave length associated with the plain wave is found after some algebraic handling:
[tex]\frac{2\pi}{\lambda} = 4[/tex]
[tex]\lambda = \frac{\pi}{2}[/tex]
The wave length of the plain wave is [tex]\frac{\pi}{2}[/tex].
c) The frequency can be calculated in term of angular frequency, that is:
[tex]f = \frac{2\pi}{\omega}[/tex]
Where [tex]f[/tex] is the frequency of the plain wave.
If [tex]\omega = 120[/tex], then:
[tex]f = \frac{2\pi}{120}[/tex]
[tex]f = \frac{\pi}{60}[/tex]
The frequency of the plain wave is [tex]\frac{\pi}{60}[/tex].
d) The speed of propagation is equal to the product of wave length and frequency:
[tex]v = \lambda \cdot f[/tex]
Given that [tex]\lambda = \frac{\pi}{2}[/tex] and [tex]f = \frac{\pi}{60}[/tex], the speed of propagation is:
[tex]v = \left(\frac{\pi}{2} \right)\cdot \left(\frac{\pi}{60} \right)[/tex]
[tex]v \approx 0.082[/tex]
The speed of propagation of the plain wave is 0.082.
