Investigating 𝒗 = 𝒇𝝀 Teacher Information Sheet Learning Outcomes Students can find the relationship between frequency and wavelength for an assumed fixed wave speed. By listening for a change in frequency when the length of the object resonating changes, students can describe how wavelength and frequency must be related-­‐ i.e. if the length resonating increases, the length of the fundamental wavelength must increase, it can be observed that the frequency decreases, so as wavelength increases, frequency decreases. Lesson timetable (all timings are estimated) 5 minutes 10-­‐15 minutes 20-­‐25 minutes 10-­‐15 minutes Settle class and take register. State aims of lesson. Recap of frequency, wavelength, pitch and wave speed. Predict the outcomes of each experiment they will carry out. Students carry out experiments ( approximately 5 minutes for each) Explain experiment outcomes Aims of the lesson 1. 2. 3. 4. Brief recap of existing knowledge of waves Hypothesise outcomes of experiments Complete experiments Explanations of experiments/summary of lesson Recap Students should already know from previous lessons that pitch is analogous to frequency, that frequency is the number of waves passing a point within a certain period of time, and that the wavelength is the distance between two identical points on the wave. x t 1 2 To demonstrate the difference between transverse and longitudinal waves we suggest the following interactive demonstration. A line of students, all holding hands send a pulse along the line. For transverse waves the person at the start of the line moves their arm up and down, passing the wave on to the next person in a similar way to a Mexican wave. For longitudinal waves, the person at the start of the line moves side to side, knocking into the next person creating a domino effect down the line. In the transverse wave, the motion of the wave is along the line, but the oscillation of the arms is up and down, i.e. the two are perpendicular. In the longitudinal wave the oscillation of the students is parallel to the direction of propagation. Another possible demonstration for explaining a transverse wave is to use a volunteer to walk steadily alongside the whiteboard with a pen in hand, moving their arm vertically (with no forward movement of the arm) up and down steadily – the movement of their hand would dictate the frequency of the waveform drawn, the walking speed signifies the wave-­‐speed, and the wavelength can then be calculated/shown. This should enable the relationship between frequency and wavelength at a fixed wave-­‐speed to be shown. By walking at the same speed but moving their arm up and down faster, a wave with a higher frequency and longer wavelength (should!) be drawn. Another way to demonstrate longitudinal waves is through Chinese whispers – the information contained in the wave is passed along the line without anyone moving. The misinterpretation of the information is analogous to interference. Predicting The students are expected to predict the outcome of the experiment before they begin. The teacher should ask the whole class a series of questions about the experiments, listed below. Students should be supplied with “traffic light” flash cards, and for each question a particular colour will represent a possible answer (red for “no”, green for “yes”, yellow for “I don’t know”). The students will then be told to make their choice and everyone should reveal their answers simultaneously, and quickly – the answers should be almost instinctive. Below is a list of questions to be asked. 1. If I decrease the length of the whistle, will the pitch increase? Yes . 2. If I increase the length of water in the test tube and blow across the top, will the pitch increase? Yes . 3. If I increase the length of water in the test tube and tap the side, will the pitch increase? No. 4. If I decrease the length of the ruler over the edge of the desk, will the pitch increase? Yes . 5. If I increase the length of a string, will the pitch increase? No . Experiments Students should then be given the worksheets and should begin the experiments. Each experiment should take no longer than 5 minutes, and students should be placed in pairs or small groups to complete the experiments. Explanations In each case we have only considered the fundamental wavelength. This is to avoid confusion with harmonics and to simplify the process and physics to ensure students understand the implications of changing wavelength on the frequency of sound produced. In the case of the string, '
slide whistle and ruler, the fundamental is half a wavelength, such that 𝑙 = , where 𝑙 is the length (
of the string, air column and ruler. If 𝑙 decreases 𝜆 must also decrease. It can be observed that reducing the length of the string results in an increase in frequency. From this students can deduce that a decrease in wavelength corresponds to an increase in frequency. What about the test tubes? In both experiments, the amount of water was increased, but tapping produced the opposite result to blowing across the top. When blowing across the test tube you are forcing the air to vibrate, so by increasing the amount of water you are decreasing the length of the air column. Decreasing the length leads to an increase in pitch, as expected. By tapping the test tube you are forcing the test tube itself to vibrate. When you add more water to the test tube, you are effectively adding mass to the test tube. A higher mass means it takes longer to return to its equilibrium position, so the frequency is reduced. Increasing the length of water leads to a decrease in the frequency. A potential analogy is using dumbbells: with no weights you can move your arms quite quickly, adding more and more weight means it takes longer to move your arm back to equilibrium, i.e. it decreases the frequency. Since frequency decreases as wavelength increases the two quantities are inversely proportional +
to one another, so 𝜆 ∝ . the constant of proportionality is the wave speed, so 𝜆 = . this can be ,
,
rearranged to the form 𝑣 = 𝑓 𝜆 . +
To show that 𝜆 ∝ set up a demonstration using a long piece of elastic string fixed between ,
two points, where one of the ends is attached to a motor to drive the string up and down. By setting the motor at the correct frequency one whole wavelength can be produced on the string. The required frequency will depend on the length of string used. If you then double the frequency, you will observe that the wavelength will half. Since the frequency is known, students can use a ruler to measure the wavelength, and hence calculate the wave speed. If the frequency is then doubled or tripled, and the corresponding wavelengths measured, students should find the wave speed to be the same for every case. Extra content This can be used if students complete the experiments quicker than anticipated, or if there is time left at the end of the lesson. In a certain material, wave speed is constant; but it isn’t the same for all materials. Sound travels faster through a solid than a liquid, and through a liquid faster than through a gas. If you place an ear on the desk and someone at the other end hits the desk, you will hear the vibration slightly sooner in the ear that is on the desk than in the other ear. The difference is very small because it is only over a short distance, but over longer distances the effect would be more apparent. If you try to talk underwater (either in the bath or at the swimming pool) it doesn’t quite sound right, because the speed has increased. One way to demonstrate that the speed of a wave is constant, but not fixed, is to use a slinky spring. Place the slinky on different surfaces, i.e. on a hard flat surface such as a table, or on carpet and send the same frequency longitudinal wave along the spring. The two will travel at different speeds. Although in this case it is the different frictional forces changing the speed it is enough to be able to see that the speed can be changed. You could also try to calculate the different wave speed by inputting a known frequency and measuring the wavelength then using 𝑣 = 𝑓𝜆. Investigating 𝒗 = 𝒇𝝀 Experiment Set-­‐‑up Sheet for Teachers • • • • • • • 5 mini experiments should be set up around the classroom. Students are expected to work in pairs or small groups depending on class size, multiples of the experiments may be required for a particularly large class. It is expected that students should spend approximately 5 minutes at each station, with every student carrying out all 5 experiments. The accompanying worksheet provided should be completed at each station. This encourages students to write down their observations. Explanations of each experiment should be discussed as a class after the experiments have all been completed. Each experiment gives the opportunity for students to investigate the relationship between frequency and wavelength; it can be assumed that the wave speed is constant. In each of the cases, only the fundamental wavelength is considered so that the fewest number of wavelengths fits in the test tube/slide whistle/ruler length. Although in theory higher harmonics could be possible, it has been simplified to this level to reduce the risk of confusion in less able students. This is a topic studied in more depth at A-­‐
Level. Mini Experiment Apparatus 1. Slide Whistle: • Students vary the length of the slide whilst blowing into it to produce a sound. • Comparisons of the change in pitch/frequency heard as the length of the slide is increased should be observed and recorded. • The relationship between length of slide and frequency should be discussed, where the longer length produces a lower pitch/frequency sound. 2. Blowing across test tubes: • A test tube rack containing 5 test tubes with varying amounts of water should be provided. • One test tube should be almost filled with water, another should be almost empty. The other three should have various amounts in between these two extremes i.e. ¼, ½, ¾ full. All test tubes should be the same length and diameter. An example of this is shown below. 1 2 3 4 5 Students are expected to blow across the top of each test tube to produce a sound, as they would with a drinks bottle or flute, and observe and record the change in pitch/frequency with the varying amounts of water. • Students should describe the relationship between the length of the air column and the frequency of the sound heard. Here the longer air column produces a lower frequency. 3. Tapping test tubes: • The same set up of 5 test tubes as in experiment 2 should be used. It is suggested that if possible food colouring could be added to this set to distinguish between the two experiments. • Students should use a wooden pencil to tap the rim of each of the test tubes in turn to cause the test tube to vibrate and produce a sound. They should again observe and record how the pitch/frequency changes with varying water levels. • Students should observe that this time the test tube with the longer water column produces a lower frequency. 4. Twanging Ruler: • A plastic/wooden ruler should be provided on the desk, students will be asked to hold the ruler in place at the edge of the desk. • By ‘twanging’ the ruler with varying length protruding over the edge of the desk, they should observe and record the change in pitch/frequency as they change the length of the ruler extended over the desk edge. • All rulers provided should be made of the same material to avoid any differences caused by varying Young’s modulus for the different material. • Students should observe that the longer the overhang, the lower the frequency of sound heard. 5. Plucking Strings: • A piece of board with 5 nails knocked into it should be provided. The nails should be approximately 4cm apart and at different distances from the edge of the board as shown in the diagram below. • • • • • Each nail should have a length of string attached, with weights on the end which is able to hang over the edge of the table. The weights help to create tension in the string. Ideally guitar strings should be used. A pencil can be placed under the strings to act as a guitar fret would. (Not shown on diagram) Students should pluck the strings and observe and record the change in pitch/frequency for the change in length of string. Students should observe that a longer string will produce a lower frequency note. Risk Assessment i. ii. iii. iv. v. vi. Risk of ruler breaking if excessive force is applied Risk of injury to feet if the weights fall off the strings or if the strings break Risk of eye damage if the strings break Risk of slipping if any water is spilled Don’t tap test tubes with unnecessary force so they smash. Don’t walk around playing the slide whistle, students could trip and choke on the whistle. Investigating 𝒗 = 𝒇𝝀 Student Worksheet Using the five experiments set up around the classroom, investigate how frequency and wavelength are related. 1. Slide Whistle: Start the experiment with the plunger/slide pulled out as long as possible. Blow into the whistle and listen to how the pitch of the sound changes as you push the plunger in. • What is vibrating? _________________________________________________________ • What is the independent variable? ___________________________________________ • What is the dependent variable? _____________________________________________ • Does the longest length vibrating have the highest or lowest pitch? _________________ Conclusion: If you decrease the length of the whistle, the frequency of the sound you hear ________________. 2. Test Tubes A: Blow across the top of each test tube to produce a note, starting with the tube with the least amount of water and continuing in order of increasing water. Listen to how the pitch of the sound changes as you play each test tube. • What is vibrating? Air, water, or water and test tube? ____________________________ • What is the independent variable? ___________________________________________ • What is the dependent variable? _____________________________________________ • Does the longest length vibrating have the highest or lowest pitch? _________________ Conclusion: If you increase the length of the ____________the frequency of the sound you hear ________________. 3. Test Tubes B: Gently tap the rim of each test tube, starting with the tube with the least amount of water and continuing in order of increasing water. Listen to how the pitch of the sound changes as you tap each test tube. • What is vibrating? Air, water, or water and test tube? ____________________________ • What is the independent variable? ___________________________________________ • What is the dependent variable? _____________________________________________ • Does the longest length vibrating have the highest or lowest pitch? _________________ Conclusion: If you increase the length of the _____________the frequency of the sound you hear ________________. 4. Ruler: Start with the ruler 25cm hanging over the edge of the desk, firmly holding it in place. Twang the ruler and listen to the pitch of the sound produced. Decrease the length of the ruler overhanging the desk in steps of 5cm, listening to the pitch of the sound produced at each step. • What is vibrating? _________________________________________________________ • What is the independent variable? ___________________________________________ • What is the dependent variable? _____________________________________________ • Does the longest length vibrating have the highest or lowest pitch? _________________ Conclusion: If you decrease the overhanging length of the ruler the frequency of the sound you hear ________________. 5. Strings: Using the board provided, pluck each string in turn starting with the shortest string. Listen to how the pitch of the note changes. • What is vibrating? _________________________________________________________ • What is the independent variable? ___________________________________________ • What is the dependent variable? _____________________________________________ • Does the longest length vibrating have the highest or lowest pitch? _________________ Conclusion: If you increase the length of the string, the frequency of the sound you hear ________________. Overall Conclusion: As the length that is vibrating increases, the frequency of the sound heard ______________. So, as wavelength increases frequency ______________.