A harmonic wave is traveling along a rope. It is observed that the oscillator that gener- ates the wave completes 38.9 in 27.4 s. Also, a given maximum travels 432 cm along the rope in 10.6 s.
What is the wavelength? Answer in units of cm.
To find the wavelength of the harmonic wave, we can use the formula:
wavelength = wave speed / frequency
We are given the time it takes for the oscillator to complete one full oscillation, but we need to find the frequency in order to use the formula. The frequency (f) is the reciprocal of the time period (T), which is the time it takes for one complete oscillation. So, we can find the frequency using the equation:
f = 1 / T
Let's calculate the frequency using the given time period:
T = 27.4 s
f = 1 / T
f = 1 / 27.4
f ≈ 0.0365 Hz
Now we need to find the wave speed. The wave speed (v) for a harmonic wave on a rope can be calculated using the equation:
v = λ * f
where λ is the wavelength. Rearranging the equation, we can solve for the wavelength:
λ = v / f
We are given the distance a given maximum (which is a wave crest or trough) travels in a certain time. The distance traveled by a wave crest or trough is equal to one full wavelength. So, we can calculate the wave speed using the given distance and time:
distance = 432 cm
time = 10.6 s
v = distance / time
v = 432 cm / 10.6 s
v ≈ 40.755 cm/s
Now we can calculate the wavelength using the formula:
λ = v / f
λ = 40.755 cm/s / 0.0365 Hz
λ ≈ 1118.493 cm
Therefore, the wavelength of the harmonic wave is approximately 1118.493 cm.