A sinusoidal wave on a string is described by the equation y = (0.191 m) sin (0.745 x - 42.9 t), where x and y are in meters and t is in seconds. If the linear mass density of the string is 11.7 g/m ...
a) ... determine the phase constant.
b) ... the phase of the wave at x = 2.29 cm and t = 0.197 s.
To determine the phase constant, we need to compare the given equation to the general equation for a sinusoidal wave:
y = A sin(kx - ωt + φ),
where A is the amplitude, k is the wave number, ω is the angular frequency, t is the time, x is the position, and φ is the phase constant.
Comparing the given equation y = (0.191 m) sin (0.745 x - 42.9 t) to the general equation, we can see that:
Amplitude (A) = 0.191 m
Wave number (k) = 0.745
Angular frequency (ω) = 42.9
Since the phase constant is not directly given, we can derive it by comparing the given equation to the general equation. The phase constant is the value that makes the equation true at time t = 0 and position x = 0.
At t = 0 and x = 0, the equation becomes:
y = (0.191 m) sin (0 - 0 + φ) = (0.191 m) sin φ,
So, y = (0.191 m) sin φ.
Now, comparing this to the given equation, we can conclude that the phase constant (φ) = 0.
a) Therefore, the phase constant is 0.
To determine the phase of the wave at x = 2.29 cm and t = 0.197 s, we need to substitute these values into the given equation:
y = (0.191 m) sin (0.745 x - 42.9 t),
Substituting x = 2.29 cm = 0.0229 m and t = 0.197 s, we get:
y = (0.191 m) sin (0.745 * 0.0229 - 42.9 * 0.197),
y = (0.191 m) sin (0.0169905 - 8.4563),
y = (0.191 m) sin (-8.4393095).
Using a calculator, we can find that sin (-8.4393095) ≈ -0.649.
b) Therefore, the phase of the wave at x = 2.29 cm and t = 0.197 s is approximately -0.649.