Students will learn how standing waves form through constructive interference and how to solve standing wave problems where either both ends are constrained or neither of the ends are constrained.
Key Equations
Standing waves for a string restricted at both ends or unrestricted at both ends
\begin{align*} f_n = \frac{n v}{2L} \; \text{ ; n is an integer} \end{align*}
Guidance
- A typical standing wave is shown below. This is the motion of a simple jump-rope. Nodes are the places where the rope doesn’t move at all; antinodes occur where the motion is greatest.
For this wave, the wavelength is . Since , the frequency of oscillation is .
- Higher harmonics can also form. Note that each end, where the rope is attached, must always be a node. Below is an example of a rope in a 5^{th} harmonic standing wave.
In general, the frequency of oscillation is , where n is the number of antinodes. The thick, dotted lines represent the wave envelope: these are the upper and lower limits to the motion of the string.
- Importantly, each of the above standing wave examples can also apply to sound waves in a closed tube, electromagnetic waves in a wire or fiber optic cable, and so on. In other words,the standing wave examples can apply to any kind of wave, as long as nodes are forced at both ends by whatever is containing/reflecting the wave back on itself.
- Resonance is a phenomenon that occurs when something that has a natural frequency of vibration (pendulum, guitar, glass, etc.) is shaken or pushed at a frequency that is equal to its natural frequency of vibration. The most dramatic example is the collapse of the Tacoma Narrows bridge due to wind causing vibrations at the bridge's natural frequency. The result is the dramatic collapse of a very large suspension bridge.
Example 1
You want to create a 4th harmonic standing wave standing wave of frequency 10 Hz on a rope. Prior to conducting your experiment, you are able to create a 1st harmonic standing wave with the exact same kind of rope at the same tension and you measure the frequency to be 3 Hz. What length of rope will you need to use in order to create your 4th harmonic standing wave at 10 Hz?
Solution
This is a two step problem. We'll start by finding how fast waves travel along the rope using the information from the prior test.
\begin{align*} f_1&=\frac{nv}{2L}\\ v&=\frac{f_12L}{n}\\ v&=\frac{3\:\text{Hz} * 2 * 1\:\text{m}}{1}\\ v&=6\text{m/s}\\ \end{align*}
Now that we have the speed of waves on the rope, we can use that to find the length we need.
\begin{align*} f_4&=\frac{nv}{2L}\\ L&=\frac{nv}{2f_4}\\ L&=\frac{4*6\:\text{m/s}}{2*10\:\text{Hz}}\\ L&=1.2\:\text{m}\\ \end{align*}
Watch this Explanation
Simulation
Wave on a String (PhET Simulation)
Time for Practice
- A violin string vibrates, when struck, as a standing wave with a frequency of \begin{align*}260\;\mathrm{Hz}\end{align*}. When you place your finger on the same string so that its length is reduced to \begin{align*}2/3\end{align*} of its original length, what is its new vibration frequency? What is the new 3rd harmonic?
- The length of the western section of the Bay Bridge is 2.7 km.
- Draw a side-view of the western section of the Bay Bridge and identify the seven nodes in this section of the bridge.
- Assume that the bridge is concrete (the speed of sound in concrete is 3200 m/s). What is the lowest frequency of vibration for the bridge? (You can assume that the towers are equally spaced, and that the central support is equidistant from both middle towers. The best way to approach this problem is by drawing in a wave that “works.”)
- What might happen if an earthquake occurs that shakes the bridge at precisely this frequency?
- The simple bridge shown here oscillated up and down pretty violently four times every second as a result of an earthquake.
- What was the frequency of the shaking in \begin{align*}Hz\end{align*}?
- Why was the bridge oscillating so violently?
- Calculate two other frequencies that would be considered “dangerous” for the bridge.
- What could you do to make the bridge safer? (Note that earthquakes rarely shake at more than \begin{align*}6 \;\mathrm{Hz}\end{align*})
Answers
- \begin{align*}390 \;\mathrm{Hz}\end{align*} , \begin{align*}1170 \;\mathrm{Hz}\end{align*}
- a. 3 wavelengths, \begin{align*}6^{th}\end{align*} harmonic b. 3.56 Hz c. Resonance will occur, the amplitude will go out of control, bridge will collapse
- a. \begin{align*}4 \;\mathrm{Hz}\end{align*} b. It was being driven near its resonant frequency. c. \begin{align*}8 \;\mathrm{Hz}, 12 \;\mathrm{Hz}\end{align*} d. Put a support in the middle, making the fundamental frequency \begin{align*}8 \;\mathrm{Hz}\end{align*}