Hard questions calculation on wave equation

Physics

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4 parts
a

0.002 m\text{0.002 m}

b

47.7 Hz\text{47.7 Hz}

c

4.19 m\text{4.19 m}

d

200 m/s\text{200 m/s}

Step-by-step solution

4 steps

Here's how to solve the problem:

The general equation for a sinusoidal wave traveling in the positive x-direction is given by:

y=Asin⁡(ωt−kx)y = A \sin(\omega t - kx)

where AA is the amplitude, ω\omega is the angular frequency, and kk is the wave number.

The given wave equation is:

y=0.002sin⁡(300t−1.5x)y = 0.002 \sin(300t - 1.5x)

Step 1: Compare the given equation with the general wave equation to identify the amplitude, angular frequency, and wave number.
From the comparison:
A=0.002 mA = 0.002 \text{ m}
ω=300 rad/s\omega = 300 \text{ rad/s}
k=1.5 rad/mk = 1.5 \text{ rad/m}

Part a· 1 step

The amplitude

  1. Determine the amplitude (AA).

    The amplitude is directly identified from the equation.

    A=0.002 mA = 0.002 \text{ m}

    The amplitude is 0.002 m\boxed{\text{0.002 m}}.

Part b

47.7 Hz\text{47.7 Hz}

The frequency

  1. Calculate the frequency (ff) using the angular frequency ω\omega.

    The relationship between angular frequency and frequency is ω=2πf\omega = 2\pi f.

    f=ω2πf = \frac{\omega}{2\pi}

    Substitute the value of ω\omega:

    f=300 rad/s2π radf = \frac{300 \text{ rad/s}}{2\pi \text{ rad}} f≈3002×3.14159 Hzf \approx \frac{300}{2 \times 3.14159} \text{ Hz} f≈47.746 Hzf \approx 47.746 \text{ Hz}

    The frequency is 47.7 Hz\boxed{\text{47.7 Hz}}.

Part c

4.19 m\text{4.19 m}

The wavelength

  1. Calculate the wavelength (λ\lambda) using the wave number kk.

    The relationship between wave number and wavelength is k=2πλk = \frac{2\pi}{\lambda}.

    λ=2πk\lambda = \frac{2\pi}{k}

    Substitute the value of kk:

    λ=2π rad1.5 rad/m\lambda = \frac{2\pi \text{ rad}}{1.5 \text{ rad/m}} λ≈2×3.141591.5 m\lambda \approx \frac{2 \times 3.14159}{1.5} \text{ m} λ≈4.189 m\lambda \approx 4.189 \text{ m}

    The wavelength is 4.19 m\boxed{\text{4.19 m}}.

Part d

200 m/s\text{200 m/s}

The speed of the wave

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

    The speed of the wave can be calculated using the formula v=ωkv = \frac{\omega}{k}.

    v=300 rad/s1.5 rad/mv = \frac{300 \text{ rad/s}}{1.5 \text{ rad/m}} v=200 m/sv = 200 \text{ m/s}

    Alternatively, using v=fλv = f\lambda:

    v=(47.746 Hz)×(4.189 m)v = (47.746 \text{ Hz}) \times (4.189 \text{ m}) v≈200 m/sv \approx 200 \text{ m/s}

    The speed of the wave is 200 m/s\boxed{\text{200 m/s}}.

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