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Zorluk: ZorWave Properties and Mathematical Wave Equation

A progressive sinusoidal wave traveling in a primary medium is governed by the equation y=0.08sin⁡(120πt−4πx)y = 0.08 \sin(120\pi t - 4\pi x), where xx and yy are measured in meters and tt in seconds. When the wave passes into a second medium, its propagation speed changes to 15 m/s15\text{ m/s}. What is the wavelength of the wave in the second medium?

  1. A
    0.50 m0.50\text{ m}
  2. 0.25 m0.25\text{ m}Cevap
  3. C
    0.125 m0.125\text{ m}
  4. D
    4.00 m4.00\text{ m}

Cevap

0.25 m0.25\text{ m}
Comparing the given equation y=0.08sin⁡(120πt−4πx)y = 0.08 \sin(120\pi t - 4\pi x) to the standard wave equation y=Asin⁡(ωt−kx)y = A \sin(\omega t - kx) gives an angular frequency ω=120π rad/s\omega = 120\pi\text{ rad/s}. The wave frequency is therefore f=120π2π=60 Hzf = \frac{120\pi}{2\pi} = 60\text{ Hz}. Since the frequency of a wave is determined solely by its source, it remains invariant when transmitting into a new medium. Given the new speed v2=15 m/sv_2 = 15\text{ m/s}, the new wavelength is λ2=v2f=1560=0.25 m\lambda_2 = \frac{v_2}{f} = \frac{15}{60} = 0.25\text{ m}.

Adım Adım Çözüm

1
Extract angular frequency (ω\omega) from the wave equation
Standard form y=Asin⁡(ωt−kx)y = A \sin(\omega t - kx) gives ω=120π rad/s\omega = 120\pi\text{ rad/s}.
The coefficient of tt in the argument of the sine function represents angular frequency.
2
Calculate the frequency of the wave in the primary medium
f=ω2π=120π2π=60 Hzf = \frac{\omega}{2\pi} = \frac{120\pi}{2\pi} = 60\text{ Hz}.
Frequency ff is related to angular frequency by ω=2πf\omega = 2\pi f.
3
Apply the principle of frequency invariance across boundary media
Frequency in the second medium f2=60 Hzf_2 = 60\text{ Hz}.
When a wave passes from one medium to another, its frequency depends only on the source and remains constant across boundaries.
4
Compute the wavelength in the second medium using the wave equation v=fλv = f\lambda
λ2=v2f2=15 m/s60 Hz=0.25 m\lambda_2 = \frac{v_2}{f_2} = \frac{15\text{ m/s}}{60\text{ Hz}} = 0.25\text{ m}.
Wavelength is inversely proportional to frequency for a given wave speed in that medium.

Anahtar Kavram

Wave Equation Parameter Extraction and Frequency Invariance across Media Boundaries
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Konular

Wave Properties and Mathematical Wave Equation

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