Maths IA · Modelling · topic 89 of 97
The pitch of a drum: radius, tension and loading · IB Maths IA idea: guide, sample graph and step-by-step plan
the head vibrates in modesStrike a drum and you hear one note, but the head is vibrating in many ways at once. You're going to test what sets the lowest note and why a small weight at the centre changes it.
- ✓ you play a drum or have one at home
- ✓ you want a model with two variables, not one
- ✓ you like logs, rates of change and checking a formula against a measurement
The big idea
A circular drumhead vibrates in modes. The lowest mode has a quoted frequency that depends on the radius a, the tension T and the mass per unit area: pitch falls as the head gets bigger and rises with the square root of the tension. Bessel functions are beyond the course, so quote the result and say where it comes from; your marks come from testing it. Measure pitch with a spectrum app, fit a power law on log-log axes, and see whether the exponents are −1 and 1/2. Then ask what a loaded centre does. Work out an effective mass for the skin by integration, predict the new pitch and compare it with the drum you hold.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Measure the lowest pitch of drums of different radius (or one drum with a sliding clamp) and fit f against a on log-log axes.
→ Your question becomes whether the gradient is −1, with an interval. Needs logarithms and a straight-line fit.Change the tuning in measured steps, with a tension proxy you can justify, and fit pitch against it.
→ You must decide what you can actually measure as tension. That decision, and its limits, is part of the exploration.Write f as a function of a and T and find how much it changes for a small change in each, holding the other fixed.
→ Rates of change with a variable held fixed. Compare the sensitivity to each and say which matters more for tuning.Add a known small mass at the centre of the head and measure the pitch drop.
→ An effective-mass integral over the mode shape. Compare the predicted drop with your measurement.Find the second, third and fourth peaks in the spectrum and divide each by the lowest.
→ The ratios are not whole numbers for a plain head. Compare them with the quoted ratios and say why a loaded drum can do better.Repeat each strike, change the striking place and compare two spectrum apps or two window lengths.
→ Your uncertainty is something you measured. It sets how firmly you can claim an exponent.Your Maths IA research question: run the check
which variable? how measured?Write your own question first. Then make sure it ticks every box:
- Names the drum, the variable you change (radius, tension or load) and how you will measure it.
- Says how pitch is read from the spectrum and what you do when several peaks appear.
- States the model you test, with its source, and which results you derive and which you quote.
- Says what you will keep fixed: the strike, the room, the microphone distance and the skin.
- Gives the number of different settings, at least seven, with repeats.
The set-up
A circular head, a rim that pulls outwards, a load at the centre and a spectrum. Draw your own drum with real measurements and say where the microphone sits.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Which drum, why you chose it, and what you predict for radius, tension and load before measuring.
The quoted formula, what each symbol means with units and where it comes from. Say plainly that the Bessel part is quoted.
How f changes with a and with T, with the other held fixed, and what the signs and sizes mean.
Your method, a sample spectrum, how you chose the lowest peak and your uncertainty.
Log-log fit of pitch against radius (or tension): gradient, interval and residuals.
The effective-mass integral, the predicted pitch and the measured one.
Back to your aim: which laws held, within what interval, and how far a loaded centre moved the pitch.
Air loading, a skin that is not uniform, a clamp that is not a perfect rim, a spectrum that is not clean. Say which limit matters most.
Where the formula, the zeros of the Bessel function and your tools came from.
What your Maths IA graph could look like
log-log: read the gradientA made-up data set to show the form of a strong graph, not the answer: pitch against radius on a log-log scale for a model drum. Yours will look different. example only
- A log-log version of this plot makes the model a straight line. Plot both if you can.
- Axis labels give units: radius in centimetres, pitch in hertz.
- Give the fitted gradient with an interval, then say whether −1 is inside it.
- Plot your repeated readings as separate dots. The spread is your uncertainty.
- Another graph to consider: the second and third peaks divided by the lowest, against your drum's tuning.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out how a drum head vibrates. Before measuring, guess what happens to the pitch if you double the radius or the tension.
- Choose the drumPick a drum you can set up in a few measured ways: sizes, tunings or a mass at the centre. Say how you will measure each.
- Quote the modelWrite the formula with units and a source. Say which part you derive and which you quote, and why.
- Trial run (1 lesson)Strike the drum five times in one setting. Read the spectrum, pick the lowest peak and see how much it varies before you plan the rest.
- Measure pitchCollect at least seven settings, with repeats, keeping the strike and microphone distance the same.
- Fit on log axesTake logs, fit a straight line and compare the gradient with the model. Give an interval and look at the residuals.
- Rates of changeFind how f changes for a small change in a, and in T, with the other held fixed. Compare the sizes.
- Go further, with a reasonAdd a centre load and predict the pitch with an effective-mass integral. HL: justify each step of the integral.
- Reflect as you goAfter every result, write one line on what it means for tuning a drum and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Power functions and logarithms – Take logs of to get a straight line, read from the gradient and give an interval.
- Rates of change with one variable fixed – Find and by ordinary differentiation, treating the other as constant.
- Concavity – Show is convex in and concave in with the second derivative, and say what this means for tuning steps.
- Definite integral for effective mass – Evaluate numerically to get the effective mass of the skin.
- Uncertainty and regression – Fit the log-log line with technology, report the gradient interval and test residuals for a pattern.
Key equations
Syllabus links
- 1.7Rational exponents and laws of logarithms
- 2.8Reciprocal and simple rational functions
- 2.9Exponential and logarithmic functions
- 5.3Differentiating powers of x
- 5.7Second derivative and graph behaviour
- 5.11Definite integrals and areas between curves
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 2.10Log scales, linearising data, log-log and semi-log graphsHL
- 4.4Correlation and linear regression
- 4.13Non-linear regression, residuals and R-squaredHL
- 5.8Trapezoidal rule
Read up first
Data and sources to start from
Strike the drum at a fixed spot, record on a phone and read the lowest spectrum peak with a spectrum app, for each radius, tuning or load.
References to cite
- Fletcher, N. H., & Rossing, T. D. (1998). The physics of musical instruments (2nd ed.). Springer.
- Rossing, T. D. (2000). Science of percussion instruments. World Scientific.
- Staacks, S., Hütz, S., Heinke, H., & Stampfer, C. (2018). Advanced tools for smartphone-based experiments: phyphox. Physics Education, 53(4), 045009.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
quote it, don't claim to derive it- Saying the model is derived when the Bessel part is only quoted. Be clear about which is which.
- Taking the loudest peak as the pitch. It may be an overtone or a room resonance.
- Calling a change in tuning a change in tension without any measurement of tension.
- Fitting on logs but reporting the gradient with no uncertainty.
- Using total mass of the skin where the formula needs mass per unit area.
- Hitting the drum in a different place each time, which changes which overtones are loud.
More Maths IA ideas like this
Tuner app: how does pitch change with string tension?●○○
Fourier series: what is in a musical note?●●●
Is Moore's law over? Fitting exponential growth●●○
The most efficient wine glass: least glass for the volume●●●
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.