CALC. A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation $y(x,t)=\left(5.60\text{ cm}\right)\sin \left[\right(0.0340\text{ rad/cm}\left)x\right]\sin \left[\right(50.0\text{ rad/s}\left)t\right]$, where the origin is at the left end of the string, the $x$-axis is along the string, and the $y$-axis is perpendicular to the string. Draw a sketch that shows the standing-wave pattern.

Verified step by step guidance

1. Understand the given wave equation: y(x, t) = (5.60 cm) sin[(0.0340 rad/cm)x] sin[(50.0 rad/s)t]. This represents a standing wave on a string.
2. Identify the harmonic: The equation describes the third harmonic. In a standing wave, the number of antinodes corresponds to the harmonic number. Therefore, there will be three antinodes.
3. Determine the wavelength: The wave number k is given as 0.0340 rad/cm. The wavelength λ is related to the wave number by the equation k = 2π/λ. Solve for λ to find the wavelength of the wave.
4. Sketch the wave: For the third harmonic, the string will have three antinodes and two nodes (excluding the endpoints). The nodes are points of zero displacement, and the antinodes are points of maximum displacement.
5. Label the sketch: Clearly mark the nodes and antinodes on the sketch. The nodes are at the fixed ends and at points along the string where the displacement is always zero. The antinodes are the points of maximum amplitude.

Key Concepts

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a wave pattern that appears to be stationary, characterized by nodes (points of no displacement) and antinodes (points of maximum displacement). In the context of a string, standing waves occur at specific frequencies known as harmonics.

Harmonics

Harmonics are the resonant frequencies at which standing waves are established on a string fixed at both ends. The fundamental frequency, or first harmonic, has one antinode and two nodes at the ends. The third harmonic, as in this problem, has three antinodes and four nodes, indicating that the string vibrates in a more complex pattern with higher frequency.

Wave Equation

The wave equation y(x, t) = (5.60 cm) sin[(0.0340 rad/cm)x] sin[(50.0 rad/s)t] describes the displacement of points on the string over time. The spatial component, sin[(0.0340 rad/cm)x], determines the wave's shape along the string, while the temporal component, sin[(50.0 rad/s)t], dictates the oscillation over time. This equation is crucial for visualizing and sketching the standing wave pattern.