Question

A wave on a string has a wave function given by \(y(x, t)=(0.0200 \mathrm{~m}) \sin \left[\left(6.35 \mathrm{~m}^{-1}\right) x+\left(2.63 \mathrm{~s}^{-1}\right) t\right]\) a) What is the amplitude of the wave? b) What is the period of the wave? c) What is the wavelength of the wave? d) What is the speed of the wave? e) In which direction does the wave travel?

Short answer

#Answer## a) Amplitude: 0.0200 m b) Period: 2.39 s c) Wavelength: 0.989 m d) Speed: 0.414 m/s e) Direction: Leftwards (negative x-direction)

Step-by-step solution

a) Amplitude of the wave

From the given wave function, we can directly read the amplitude as it is the coefficient of the sine term. The amplitude \(A=0.0200\,\mathrm{m}\).

b) Period of the wave

From the given wave function, we can see that the angular frequency \(\omega = 2.63 \mathrm{~s}^{-1}\). To find the period of the wave, we can use the formula: \(T=\frac{2\pi}{\omega}\). So, \(T=\frac{2\pi}{2.63\mathrm{~s}^{-1}} = 2.39\,\mathrm{s}\).

c) Wavelength of the wave

From the given wave function, we can see that the wave number \(k = 6.35\mathrm{~m}^{-1}\). To find the wavelength of the wave, we can use the formula: \(\lambda = \frac{2\pi}{k}\). So, \(\lambda = \frac{2\pi}{6.35 \mathrm{~m}^{-1}} = 0.989\,\mathrm{m}\).

d) Speed of the wave

We can find the speed of the wave using the formula: \(v = \frac{\omega}{k}\). So, \(v = \frac{2.63\mathrm{~s}^{-1}}{6.35\mathrm{~m}^{-1}} = 0.414\,\mathrm{m/s}\).

e) Direction of the wave

From the given wave function, the term inside the sine function is \([6.35\mathrm{~m^{-1}} x + 2.63\mathrm{~s^{-1}}t]\) which can be matched to \([kx \pm \omega t]\) of general wave function. As we see that it has a plus sign in between \(kx\) and \(\omega t\), the wave is propagating in the negative x-direction (leftwards).

Key concepts

Amplitude of a Wave

Imagine a wave on a string moving up and down as it travels through space. The amplitude of a wave is the maximum distance that the particles of the medium (in this case, the string) move from their rest position as the wave passes. It determines the wave's energy; a higher amplitude means more energy. In our example, the amplitude is directly stated as 0.0200 meters. This value represents the peak height of the wave above or below the equilibrium position. The amplitude is crucial as it affects the loudness in sound waves or the brightness in light waves.

Understanding amplitude is essential for interpreting the physical effects of waves, for instance, how strongly an earthquake shakes the ground or how intense a radio signal is received.

Period of a Wave

The period of a wave is the time it takes for one complete cycle of the wave to pass a given point. It reflects how frequently the wave cycles repeat and is measured in seconds. The period is the inverse of frequency, meaning waves with a higher frequency have shorter periods. To calculate the period \(T\) from the angular frequency \(\omega\), we use \(T=\frac{2\pi}{\omega}\). For our wave on a string, with \(\omega=2.63 \mathrm{~s^{-1}}\), the period is \(2.39\,\mathrm{s}\).

The period is particularly significant when dealing with phenomena that oscillate or repeat at regular intervals, like pendulums in clocks or the orbits of planets, helping us to predict their motion over time.

Wavelength of a Wave

The distance between two corresponding points on consecutive cycles of a wave, such as crest to crest or trough to trough, is the wavelength of a wave. It is represented by the Greek letter \(\lambda\) and typically measured in meters. Wavelength determines the type of radiation (e.g., radio, visible light, or X-rays) for electromagnetic waves. You calculate it by \(\lambda = \frac{2\pi}{k}\), where \(k\) is the wave number. With a wave number of \(6.35\,\mathrm{m^{-1}}\) in our example, the wavelength \(\lambda\) comes out to be \(0.989\,\mathrm{m}\).

Knowing the wavelength helps in understanding how waves will behave when encountering obstacles; for example, a wave with a long wavelength can bend around objects, a phenomenon known as diffraction.

Speed of a Wave

The speed of a wave is the rate at which the wave travels through the medium; it's how fast the wave's energy is transmitted from one place to another. For waves on a string, the speed depends on the tension and mass per unit length of the string. In physics, we calculate this speed using \(v = \frac{\omega}{k}\), which combines the wave's angular frequency \(\omega\) and wave number \(k\). Our string wave travels at \(0.414\,\mathrm{m/s}\), a relatively moderate pace revealing how quickly disturbances are conveyed along the string.

The speed of a wave is constant for a given medium under consistent conditions, but changes when the wave moves from one medium to another, leading to the bending of the wave's path, known as refraction.

Wave Propagation Direction

The wave propagation direction is the direction in which the wave energy moves. For a string wave, we can deduce this from the wave function's structure. A positive coefficient in front of the time variable (\(t\)) usually means the wave moves in the positive direction; if negative, it moves in the opposite direction. Our wave function contained a positive time term, indicating the wave travels leftwards, or in the negative x-direction.

Understanding the direction of wave propagation is key for applications such as forecasting wave paths in the ocean, designing antennas for optimal signal transmission, or even in medical imaging techniques like ultrasound.