Question

The displacement from equilibrium caused by a wave on a string is given by \(y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\right.\) \(\left.\left(800 . \mathrm{s}^{-1}\right) t\right] .\) For this wave, what are the (a) amplitude, (b) number of waves in \(1.00 \mathrm{~m},\) (c) number of complete cycles in \(1.00 \mathrm{~s},\) (d) wavelength, and (e) speed?

Short answer

Answer: The amplitude is \(0.00200m\), the number of waves in 1.00m is 40, there are approximately 127.32 complete cycles in 1.00s, the wavelength is approximately \(0.15708m\), and the wave speed is approximately \(20m/s\).

Step-by-step solution

(a) Amplitude

To find the amplitude, compare the given wave equation with the general equation of a sinusoidal wave. The amplitude \(A\) is the coefficient of the sine function. In this case, it is \(-0.00200 m\). The amplitude is always a positive value, so the amplitude of this wave is \(0.00200m\).

(b) Number of Waves in 1.00m

To find the number of waves in 1.00m, we need the wave number \(k\). The wave number is found by looking at the coefficient of the \(x\) term in the sine function. In this case, the wave number \(k = 40.0 m^{-1}\). The number of waves in 1.00m is equal to the wave number, which in this case is 40.

(c) Number of Complete Cycles in 1.00s

To find the number of complete cycles in 1.00s, we need the angular frequency \(\omega\). The angular frequency is found by looking at the coefficient of the \(t\) term in the sine function. In this case, the angular frequency \(\omega = 800 s^{-1}\). To find the number of cycles, we need to convert angular frequency to frequency (\(f\)). The relationship between angular frequency and frequency is given by \(\omega = 2\pi f\). Therefore, we have \(f = \frac{\omega}{2\pi} = \frac{800}{2\pi} \approx 127.32 Hz\). This means there are 127.32 complete cycles in 1.00s.

(d) Wavelength

To find the wavelength, we need to use the relationship between wavelength and wave number, which is given by \(\lambda = \frac{2\pi}{k}\). Using the given wave number \(k = 40.0 m^{-1}\), we can find the wavelength as \(\lambda = \frac{2\pi}{40} = \frac{\pi}{20} \approx 0.15708m\).

(e) Wave Speed

To find the wave speed, we use the relationship between speed (\(v\)), frequency (\(f\)), and wavelength (\(\lambda\)) given by \(v = f\lambda\). We have the frequency \(f \approx 127.32 Hz\) and wavelength \(\lambda \approx 0.15708m\). So, the wave speed is \(v \approx 127.32 \times 0.15708 \approx 20m/s\).

Key concepts

Amplitude

When we talk about waves, the amplitude refers to the maximum displacement from the equilibrium position. In simple terms, it tells us how "tall" or "deep" the wave is. For the wave given by the equation \( y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[(40.0 \mathrm{~m}^{-1}) x - (800 \mathrm{s}^{-1}) t \right] \), the amplitude is found by looking at the coefficient before the sine function.
This coefficient is \(-0.00200 \mathrm{~m}\). Amplitude is always expressed as a positive value, representing the absolute maximum height of the wave from its rest position. Thus, the amplitude for this wave is \(0.00200 \mathrm{~m}\).

• Amplitude is not affected by the wave's speed or direction of movement.
• Even if a wave is moving left or right, its amplitude remains constant as long as the wave doesn't dissipate.
• The larger the amplitude, the more energy the wave carries.

Wavelength

The wavelength of a wave is the distance over which the wave's shape repeats. It is the distance between two corresponding parts of the wave, such as two consecutive crests or troughs. Wavelength is crucial in understanding how waves interact with the environment.
In our example, the wave number \( k \) is given as \( 40.0 \mathrm{~m}^{-1} \). The wave number is directly related to the wavelength by the formula \( \lambda = \frac{2\pi}{k} \).
By substituting the given wave number, we calculate the wavelength:
\[ \lambda = \frac{2\pi}{40} = \frac{\pi}{20} \approx 0.15708 \mathrm{~m} \]

• Wavelength is crucial for determining wave interference patterns and diffraction.
• Longer wavelengths mean lower frequencies, given a constant wave speed.
• In a ripple on a pond, the wavelength is the distance between waves that you can see.

Frequency

Frequency describes how often the wave oscillates per unit time. It tells us how many cycles or waves pass through a point every second and is measured in hertz (Hz). For sinusoidal waves, the angular frequency \( \omega \) is related to the frequency by \( \omega = 2\pi f \).
From the wave equation, \( \omega = 800 \mathrm{s}^{-1} \). By using the relationship, we can find the frequency:
\[ f = \frac{\omega}{2\pi} = \frac{800}{2\pi} \approx 127.32 \mathrm{~Hz} \]

• A higher frequency means more wave cycles per second.
• In sound waves, higher frequencies are perceived as higher pitches.
• Frequency is inversely related to wavelength in a given wave speed.

Wave Speed

Wave speed refers to how fast the wave travels through the medium. It can be found using the relationship \( v = f\lambda \), where \( v \) stands for wave speed, \( f \) for frequency, and \( \lambda \) for wavelength. In our wave equation, using previously calculated values:
\( f \approx 127.32 \mathrm{~Hz} \) and \( \lambda \approx 0.15708 \mathrm{~m} \).
Substituting these results gives:
\[ v \approx 127.32 \times 0.15708 \approx 20 \mathrm{~m/s} \]

• Wave speed is constant for a given medium and is independent of the frequency or wavelength alone.
• In water, light, or sound waves, speed indicates how quickly energy is transferred.
• The speed of sound in air is about \( 343 \mathrm{~m/s} \), while in water, it is faster due to denser particles.