To calculate the various properties of the wave described by the equation f(x, t) = a sin (tbx + qt), let's break down the given values:
a = 30.0 mm = 30.0 × 10^-3 m (convert mm to m)
b = 0.3 m
q = 10.5 s
Now, let's find the answers step by step:
(a) Amplitude:
The amplitude of the wave is given by the value of "a" in the equation. So, the amplitude is 30.0 × 10^-3 m.
(b) Wavelength:
The wavelength (λ) of a wave is given by the formula λ = 2π / b. Plugging in the value of "b" into the formula, we get:
λ = 2π / 0.3 = 6.28 m
(c) Period:
The period (T) of a wave is the time it takes for one complete cycle. It is given by the formula T = 2π / q. Substituting the value of "q" into the formula, we have:
T = 2π / 10.5 = 0.6 s (approximately)
(d) Speed of the wave:
The speed of a wave (v) is calculated by multiplying the wavelength (λ) by the frequency (f), where frequency (f) is the reciprocal of the period (T). So, v = λ * f.
In this case, since f = 1 / T, we can substitute T = 0.6 s to find:
v = λ * (1 / T) = 6.28 m * (1 / 0.6 s) = 10.47 m/s (approximately)
(e) Displacement at z=0.5 m and t = 1.60 s:
To compute the displacement at a specific point and time, we can substitute the given values into the equation f(x, t) = a sin (tbx + qt).
Here, z = 0.5 m and t = 1.60 s. Plugging these values in, we get:
f(x, t) = a sin (tbx + qt)
f(x, t) = 0.03 * sin (0.3 * 0.5 + 10.5 * 1.60)
Now, you can calculate the numerical value for the displacement by evaluating the equation.