Question

The equation of a transverse wave traveling along a string is given by $$ y=(2.30 \mathrm{~mm}) \sin [(1822 \mathrm{rad} / \mathrm{m}) x-(588 \mathrm{rad} / \mathrm{s}) t] $$ Find \((a)\) the amplitude, \((b)\) the frequency, \((c)\) the velocity, \((d)\) the wavelength of the wave, and ( \(e\) ) the maximum transverse speed of a particle in the string.

Short answer

The amplitude of the wave is 2.30mm. The frequency is \( 588 / 2\pi Hz \). The velocity is \( 588 / 1822 m/s \). Wavelength is \( 2\pi / 1822 m \). The maximum transverse speed is 1351.4 mm/s.

Step-by-step solution

Amplitude

The amplitude of the wave is directly given in the equation as 2.30mm.

Frequency

We can calculate the frequency by using the relation \( \omega = 2\pi f \), where \(\omega \) is given by \( 588 rad/s \). Therefore, \( f = \omega / 2\pi = 588 / 2\pi Hz \).

Velocity

The velocity can be calculated by the relation \( v = \omega / k \), where \( \omega \) is given by \( 588 rad/s \) and \( k = 1822 rad/m \). Therefore, \( v = 588 / 1822 m/s \).

Wavelength

Wavelength can be found from the relation \( \lambda = 2\pi / k \), where \( k = 1822 rad/m \). Therefore, \( \lambda = 2\pi / 1822 m \).

Maximum Transverse Speed

The maximum transverse speed is given by the product of amplitude and angular frequency, \( v_{max} = A\omega = 2.30mm \times 588 rad/s = 1351.4 mm/s \).

Key concepts

Wave Amplitude

The wave amplitude represents the maximum extent of a vibration or oscillation, measured from the position of equilibrium. Amplitude is a critical parameter because it is directly related to the energy carried by the wave; higher amplitudes mean more energy. In the context of the given transverse wave equation, the amplitude can be easily identified as the coefficient in front of the sine function, which is 2.30 mm in this case. This means that the particles in the string move up and down by a maximum of 2.30 mm from their rest position as the wave passes through them.

Wave Frequency

Wave frequency is defined as the number of cycles a wave completes per unit time and is measured in hertz (Hz). It determines how fast the wave oscillates. In the given equation, the frequency isn't directly stated, but we can find it by using the angular frequency \(\omega\), which is related to the physical frequency by the equation \(f = \frac{\omega}{2\pi}\). Here, with \(\omega = 588\) rad/s, the frequency is calculated to be approximately \(\frac{588}{2\pi}\) Hz, which gives us the number of times the wave oscillates up and down each second.

Wave Velocity

Wave velocity is the speed at which the wave travels through the medium. It's a fundamental aspect of wave behavior as it affects how quickly the energy is transmitted. From the equation provided, we use the relation \(v = \frac{\omega}{k}\), where \(\omega\) is the angular frequency and \(k\) is the wave number. In this particular problem, with \(\omega = 588\) rad/s and \(k = 1822\) rad/m, the wave velocity is \(\frac{588}{1822}\) m/s, which tells us how fast the wave pattern is propagating along the string.

Wavelength

Wavelength is the spatial period of the wave, the distance over which the wave's shape repeats. It is inversely related to the wave number \(k\) by the relation \(\lambda = \frac{2\pi}{k}\), where \(\lambda\) is the wavelength. For our specific situation, using \(k = 1822\) rad/m yields a wavelength of \(\frac{2\pi}{1822}\) m. This physical quantity is essential for understanding wave interactions like interference and diffraction, and in this context, it signifies the length of one complete cycle of the wave along the string.

Transverse Speed

Transverse speed refers to the peak speed at which points on the string move up and down as the wave travels through it. It's an important concept for applications involving wave energy and resonance. The maximum transverse speed, as derived in the exercise, is calculated by multiplying the amplitude by the angular frequency, resulting in \(v_{max} = A\omega = 2.30\) mm \(\times\) 588 rad/s or approximately 1351.4 mm/s. This represents the fastest movement that any point on the string will perform due to the wave, crucial in determining the forces and stresses that might affect the string.