RaySix

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Wave Properties and Calculations

Created by @nathanedwards

at November 1st 2023, 5:21:37 am.

AP_Physics_1-Questions

#frequency

#wavelength

#wave

AP Physics 1 Exam Question:

A wave is described by the equation:

y(x, t) = 2 \sin((20 \pi \, \text{m}^{-1})x - (100 \pi \, \text{s}^{-1})t)

where $y$ is the displacement of the wave at position $x$ and time $t$ .

a) What is the wavelength of the wave?

b) What is the frequency of the wave?

c) What is the maximum displacement of the wave?

d) What is the speed of the wave?

Answer:

a) The wavelength of a wave is the distance between two consecutive points in the wave that are in the same phase. In the given equation, the coefficient of $x$ is $20 \pi \, \text{m}^{-1}$ . Since $2 \pi$ radians corresponds to one complete cycle, we can find the wavelength using the equation:

\text{wavelength} = \frac{{2 \pi}}{{\text{wave number}}}

where wave number $k = 20 \pi \, \text{m}^{-1}$ .

\text{wavelength} = \frac{{2 \pi}}{{20 \pi \, \text{m}^{-1}}} = \frac{1}{10} \, \text{m}

Therefore, the wavelength of the wave is $\frac{1}{10} \, \text{m}$ .

b) The frequency of a wave is the number of complete cycles passing a point in one second. It is defined by the equation:

\text{frequency} = \frac{{\text{speed of the wave}}}{{\text{wavelength}}}

The speed of the wave is the coefficient of $t$ in the given equation, which is $100 \pi \, \text{s}^{-1}$ .

\text{frequency} = \frac{{100 \pi \, \text{s}^{-1}}}{{\frac{1}{10} \, \text{m}}} = 1000 \, \text{Hz}

Therefore, the frequency of the wave is $1000 \, \text{Hz}$ .

c) The maximum displacement of the wave is the amplitude, which is the coefficient of the sine function in the given equation. In this case, the maximum displacement is $2$ .

Therefore, the maximum displacement of the wave is $2 \, \text{m}$ .

d) The speed of a wave is given by the equation:

\text{speed of the wave} = \text{wavelength} \times \text{frequency}

Using the values we calculated earlier, we have:

\text{speed of the wave} = \left(\frac{1}{10} \, \text{m}\right) \times (1000 \, \text{Hz}) = 100 \, \text{m/s}

Therefore, the speed of the wave is $100 \, \text{m/s}$ .