Question

A steel cable consists of two sections with different cross-sectional areas, \(A_{1}\) and \(A_{2}\). A sinusoidal traveling wave is sent down this cable from the thin end of the cable. What happens to the wave on encountering the \(A_{1} / A_{2}\) boundary? How do the speed, frequency and wavelength of the wave change?

Short answer

Answer: When a sinusoidal traveling wave encounters the boundary between two sections of a steel cable with different cross-sectional areas, the speed of the wave changes in accordance with the ratio of the cross-sectional areas, and the wavelength changes in the same proportion as the wave speed. The frequency of the wave remains constant throughout the propagation.

Step-by-step solution

Understand the relationship between wave speed and cable properties

In a steel cable, the wave speed in each section is given by \(v = \sqrt{\frac{T}{\mu}}\), where \(T\) is the tension in the cable and \(\mu\) is the linear mass density of the cable. The linear mass density depends on the cross-sectional area: \(\mu = \frac{M}{L} = \frac{\rho V}{L} = \rho A\), where \(M\) is the mass of the cable, \(\rho\) is the material's density, \(V\) is the volume of the cable, \(L\) is the length of the cable, and \(A\) is the cross-sectional area. So the wave speed can be rewritten as \(v = \sqrt{\frac{T}{\rho A}}\). This equation shows that the wave speed in each section depends inversely on the square root of the respective cross-sectional area.

Analyze changes in wave speed

When the wave reaches the boundary between the two sections, the speed of the wave changes due to the difference in the cross-sectional areas. If we denote the wave speeds in the two sections as \(v_{1}\) and \(v_{2}\), we have \(v_{1} = \sqrt{\frac{T}{\rho A_{1}}}\) and \(v_{2} = \sqrt{\frac{T}{\rho A_{2}}}\). As the tension remains constant across the boundary, the ratio of the wave speeds in the two sections is given by \(\frac{v_{1}}{v_{2}} = \sqrt{\frac{A_{2}}{A_{1}}}\).

Evaluate changes in frequency

The frequency of the wave remains unchanged across the boundary since the frequency is determined by the source of the wave and remains constant throughout its propagation. Therefore, the frequency in both sections is the same: \(f_{1} = f_{2}\).

Determine changes in wavelength

Now we need to find the change in the wavelength of the wave. The relationship between wave speed, frequency, and wavelength is given by \(v = f\lambda\), where \(\lambda\) is the wavelength. For the two sections, we have \(v_{1} = f_{1}\lambda_{1}\) and \(v_{2} = f_{2}\lambda_{2}\). Since the frequency is the same in both sections, we can form a ratio of the wavelengths: \(\frac{\lambda_{1}}{\lambda_{2}} = \frac{v_{1}}{v_{2}} = \sqrt{\frac{A_{2}}{A_{1}}}\).

Summary

When a sinusoidal traveling wave encounters the boundary between two sections of a steel cable with different cross-sectional areas, the speed of the wave changes in accordance with \(\frac{v_{1}}{v_{2}} = \sqrt{\frac{A_{2}}{A_{1}}}\). The frequency of the wave remains constant, while the wavelength changes in the same proportion as the wave speed: \(\frac{\lambda_{1}}{\lambda_{2}} = \sqrt{\frac{A_{2}}{A_{1}}}\).

Key concepts

Wave Propagation

When we talk about wave propagation, we're referring to how waves travel through different mediums. In the case of a sinusoidal traveling wave in a steel cable, the medium is the cable itself. The wave follows principles that dictate how any disturbance moves through a material. Interestingly, when a wave encounters a boundary - here between two different cross-sectional areas of the cable - it doesn't just stop; it changes its properties.

As the wave moves from the section with area A₁ to the section with A₂, its speed alters due to the differing characteristics of the medium, which is directly tied into the wave's energy and the medium's ability to transmit that energy. This concept is particularly relevant when considering real-world applications such as communication technologies or seismic wave analysis, where understanding wave propagation can mean the difference between clear signals and lost information or between safe buildings and structural failures.

Wave Speed Relationship

The wave speed relationship is a fundamental aspect of understanding waves. It tells us how fast a wave travels through a medium and is dictated by the medium's properties. For our sinusoidal wave traveling through a steel cable, the speed depends on the tension in the cable and the linear mass density, or mass per unit length, of the cable.

The formula we work with is v = \(\sqrt{\frac{T}{\mu}}\), showing us that wave speed is inversely proportional to the square root of the cross-sectional area. Essentially, if you encounter a section of the cable with a larger cross-sectional area, you can expect the wave to slow down because the wave has more material to move through. Understanding this relationship is crucial for engineers and physicists who design materials and structures that must accommodate or harness waves, such as bridges or optical fibers.

Frequency of a Wave

Frequency is the heartbeat of a wave. It's defined as the number of complete oscillations that a wave undergoes per unit of time and is measured in hertz (Hz). When we say that the frequency of the sinusoidal wave remains unchanged as it crosses the boundary in the steel cable, it's like saying the heart rate of the wave is constant.

It's the source of the wave that determines this constant frequency, not the medium it travels through. So no matter how much the wave may stretch or compress in terms of wavelength when meeting a new medium, the frequency set by the source isn't affected. This is a crucial concept in applications like musical instruments manufacturing, where the frequency translates to pitch, or in telecommunications, where frequency stability is key to reliable signal transmission.

Wavelength Change

Wavelength is the distance between two consecutive peaks (or troughs) of a wave and is inversely related to frequency. As the wave crosses the boundary within the cable from a thinner area to a thicker one, while the frequency remains fixed, the speed and wavelength must adjust accordingly. The relationship v = fλ becomes essential here, as it connects wave speed, frequency, and wavelength.

Understanding that the wave will undergo a wavelength change is important not just in physics classes but also in real-life scenarios such as optics and acoustics, where the physical properties of lenses or concert halls are adjusted to optimize the wavelengths for the best performance. The wavelength affects how the wave carries energy and information, so manipulating the wavelength can control how effective a wave-based technology will be.