A uniform string of linear mass density \(\lambda_{0}\) and under a tension \(\tau_{0}\) has a small bead of mass \(m\) attached to it at \(x=0\). Find expressions for the complex amplitude and the power reflection and transmission coefficients for sinusoidal waves brought about by the mass discontinuity at the origin. Do these coefficients hold for a wave of arbitrary shape?

Wave phenomena can be effectively described using the concept of complex amplitude. This approach simplifies calculations and allows for a deeper understanding of wave behavior, especially when dealing with interactions at boundaries. In the context of a uniform string, as presented in the textbook problem, the complex amplitude represents the sinusoidal waves' magnitude and phase.
When sinusoidal waves meet a discontinuity, such as a mass hanging on a string, we can describe each side of the discontinuity using the sum of forward and backward traveling waves. This is done by applying Euler's formula, which relates complex exponentials to sines and cosines. The forward and backward traveling waves can be represented as:

• \forall x < 0, \(y_{LEFT}(x,t) = A_{LEFT}e^{j(kx-\textstyle\omega t)}+B_{LEFT}e^{-j(kx+\textstyle\omega t)}\)
• \forall x > 0, \(y_{RIGHT}(x,t) = A_{RIGHT}e^{j(kx-\textstyle\omega t)}+B_{RIGHT}e^{-j(kx+\textstyle\omega t)}\)

In these expressions, \(j\) is the imaginary unit, \(k\) is the wave number which relates to the wave's spatial frequency, and \(\textstyle\omega\) is the angular frequency related to the wave's temporal frequency. The coefficients \(A_{LEFT}\) and \(B_{LEFT}\) represent the amplitude of the forward and backward traveling waves on the left side of the string, while \(A_{RIGHT}\) and \(B_{RIGHT}\) represent the same for the right side.

In wave mechanics, boundary conditions are critical for determining how waves behave when encountering obstacles or changes in medium properties. The textbook example requires applying boundary conditions at the point where a mass is attached to a string. The continuity of displacement at this point dictates that the displacement on either side of the mass must be equal, leading to the condition:\(y_{LEFT}(0,t) = y_{RIGHT}(0,t)\).
Additionally, considering the forces involved, the change in tension across the mass must compensate for its inertia, which introduces a relation involving the tension in the string, the changes in the slopes of the waves on both sides, and the acceleration of the mass:\(\tau_0\frac{\partial y_{LEFT}(0,t)}{\partial x}-\tau_0\frac{\partial y_{RIGHT}(0,t)}{\partial x}=m\frac{\partial^2 y}{\partial t^2}(0,t)\).
These boundary conditions allow us to solve for the complex amplitudes of the reflected and transmitted waves, which are embedded within the solutions for \(B_{LEFT}\) and \(A_{RIGHT}\). By connecting these conditions with the wave equations, we can gain insights into the behavior of waves at boundaries and predict outcomes such as reflection, transmission, and even standing wave patterns.

Linear mass density, symbolized as \(\lambda_0\), is a property of a string that relates to waves in a fundamental way. It affects how fast waves travel on the string and is defined as the mass per unit length of the string. In the context of the exercise, the string’s tension \(\tau_0\) and linear mass density \(\lambda_0\) determine the wave speed through the relationship:\(v = \sqrt{\frac{\tau_0}{\lambda_0}}\).
Understanding the role of linear mass density is essential in predicting wave motion. A string with a higher linear mass density will have slower wave speeds compared to a string with a lighter mass density under the same tension. The linear mass density also influences the reflection and transmission of waves. In the case where a mass is attached to a string, the local linear mass density effectively changes, altering the behavior of waves at that point. Variations in linear mass density are central to the design of musical instruments and other systems where control over wave propagation is essential.