Fixed Boundary Conditions and Standing Waves §

Introduction §

In the study of waves and vibrations, understanding fixed boundary conditions and standing waves is crucial. This note will delve into the concepts of fixed boundary conditions and how they lead to the formation of standing waves, particularly in the context of undergraduate physics and mathematics.

Fixed Boundary Conditions §

Fixed boundary conditions occur when a wave encounters a boundary that it cannot displace. This typically happens at the endpoints of a medium, like a string tied at both ends. When a wave reaches these fixed points, it undergoes a reflection with a phase shift of $180^{\circ }$, also known as an inversion.

Mathematical Representation §

Mathematically, a fixed boundary at $x=0$ can be represented as: $y(0,t)=0$ This indicates that at the point $x=0$, no matter what time $t$ it is, the displacement $y$ is always zero, signifying a fixed point.

Standing Waves §

Standing waves, also known as stationary waves, are a result of the interference of two waves of the same frequency and amplitude traveling in opposite directions. They are characterized by nodes and antinodes.

Nodes and Antinodes §

• Nodes: Points of zero amplitude, occurring at fixed boundaries.
• Antinodes: Points of maximum amplitude, occurring midway between nodes.

Formation §

Standing waves are formed when an incident wave and its reflected wave (due to fixed boundary conditions) superimpose. The interference can be constructive (in-phase) or destructive (out-of-phase), leading to the characteristic pattern of nodes and antinodes.

Mathematical Description §

The mathematical form of a standing wave on a string of length $L$ can be described by: $y(x,t)=2A\sin (kx)\cos (\omega t)$ where $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency.

Standing Waves in a String with Fixed Ends §

In a string with both ends fixed, standing waves occur at specific frequencies known as harmonic frequencies. The fundamental frequency (first harmonic) is the lowest frequency at which a standing wave can form.

Harmonic Frequencies §

The $n$th harmonic frequency is given by: $f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu }}$ where $T$ is the tension in the string, \mu) is the linear mass density, and $n$ is the harmonic number (an integer).

Conclusion and Test Questions §

Understanding fixed boundary conditions and standing waves is fundamental in physics, especially in acoustics and the study of vibrations.

Test Questions §

1. Question: What happens to a wave when it encounters a fixed boundary? Back: The wave reflects with a phase shift of $180^{\circ }$.
2. Question: Describe the difference between nodes and antinodes in a standing wave. Back: Nodes are points of zero amplitude, while antinodes are points of maximum amplitude.
3. Question: How is the frequency of the third harmonic related to the fundamental frequency in a string fixed at both ends? Back: The third harmonic frequency is three times the fundamental frequency.