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Given y = A sin(2000t - (pi/16)x), calculate wavelength, frequency, and speed of the wave.

Physics

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The given equation for a plane progressive wave is:

y=Asin⁡(2000t−π16x)y = A \sin\left(2000t - \frac{\pi}{16}x\right)

This equation is in the standard form y=Asin⁡(ωt−kx)y = A \sin(\omega t - kx), where:

  • AA is the amplitude
  • ω\omega is the angular frequency
  • kk is the wave number

From the given equation, we can identify:
Angular frequency, ω=2000\omega = 2000 rad/s
Wave number, k=π16k = \frac{\pi}{16} rad/m

Now, let's calculate the required quantities.

  1. Calculate the wavelength (λ\lambda).

    The wave number kk is related to the wavelength λ\lambda by the formula k=2πλk = \frac{2\pi}{\lambda}.
    Rearranging for λ\lambda:

    λ=2πk\lambda = \frac{2\pi}{k}

    Substitute the value of kk:

    λ=2ππ16\lambda = \frac{2\pi}{\frac{\pi}{16}} λ=2π×16π\lambda = 2\pi \times \frac{16}{\pi} λ=32 m\lambda = 32\, \text{m}

    The wavelength is 32 m\boxed{\text{32 m}}.

  2. Calculate the frequency (ff).

    The angular frequency ω\omega is related to the frequency ff by the formula ω=2πf\omega = 2\pi f.
    Rearranging for ff:

    f=ω2πf = \frac{\omega}{2\pi}

    Substitute the value of ω\omega:

    f=20002πf = \frac{2000}{2\pi} f=1000π Hzf = \frac{1000}{\pi}\, \text{Hz}

    Using π≈3.142\pi \approx 3.142:

    f≈10003.142≈318.3 Hzf \approx \frac{1000}{3.142} \approx 318.3\, \text{Hz}

    The frequency is 318.3 Hz\boxed{\text{318.3 Hz}}.

  3. Calculate the speed (vv) of the wave.

    The speed of the wave vv can be calculated using the formula v=ωkv = \frac{\omega}{k}.
    Substitute the values of ω\omega and kk:

    v=2000π16v = \frac{2000}{\frac{\pi}{16}} v=2000×16πv = 2000 \times \frac{16}{\pi} v=32000π m/sv = \frac{32000}{\pi}\, \text{m/s}

    Using π≈3.142\pi \approx 3.142:

    v≈320003.142≈10184.6 m/sv \approx \frac{32000}{3.142} \approx 10184.6\, \text{m/s}

    The speed of the wave is 10184.6 m/s\boxed{\text{10184.6 m/s}}.

  4. Calculate the period (TT) of the wave.

    The period TT is the reciprocal of the frequency ff:

    T=1fT = \frac{1}{f}

    Substitute the exact value of ff:

    T=11000πT = \frac{1}{\frac{1000}{\pi}} T=π1000 sT = \frac{\pi}{1000}\, \text{s}

    Using π≈3.142\pi \approx 3.142:

    T≈3.1421000≈0.003142 sT \approx \frac{3.142}{1000} \approx 0.003142\, \text{s}

    The period is 0.003142 s\boxed{\text{0.003142 s}}.

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