Mechanical waves and sound: Year 4 physics

In mid-September 2026 Year 4 physics at the liceo scientifico (the Italian science-focused high school) has just restarted, and the first topic of the term is mechanical waves and sound: interference, standing waves, decibels and the Doppler effect. The first test arrives in five to eight weeks, so between the end of October and mid-November.

At the science high school, Year 4 physics has three hours a week, one more than at the classical or human-sciences high schools, and it shows in the tests: the problems are longer, often in several steps (first find an intermediate quantity, then use it for the real question), and the theory is sometimes backed up with a lab experiment — a vibrating string, a resonance tube, a wave generator in a ripple tank — before the written test. The chapter also weighs forwards: what is consolidated here on waves, speed of propagation and superposition is the direct basis of electromagnetic induction and electromagnetic waves in Year 5, where the same relation $v = λ f$ comes back applied to light.

There is a scheduling problem worth pointing out right away: the formula for sound intensity level in decibels uses a logarithm, but in maths exponentials and logarithms only arrive in January-February. The physics teacher introduces the base-10 logarithm from scratch, in two lines, before the full theory has been done in class.

The quantities of a wave and superposition

A periodic wave is described by wavelength $λ$ , frequency $f$ and speed of propagation $v$ , linked by

v = λ f

The speed depends on the medium — the stiffer the medium, the faster a mechanical wave propagates — and not on the frequency: changing the frequency of a source does not change the speed of the wave in the same medium, it only changes the wavelength.

When two waves meet they superpose, and it is their path difference $Δ r$ — how much longer one route is than the other to reach the same point — that decides whether the interference is constructive or destructive:

Δ r = nλ (constructive), Δ r = (n + \frac{1}{2}) λ (destructive), n = 0, 1, 2, \dots

If the two waves start in phase, a path difference of a whole number of wavelengths puts them back in phase (they reinforce each other); half a wavelength puts them in antiphase (they cancel).

Standing waves: the harmonics of a string

On a string fixed at both ends, standing waves form only for certain wavelengths, those for which the string “closes” a whole number of half-wavelengths:

L = n \frac{λ _{n}}{2} ⟹ λ_{n} = \frac{2 L}{n}, f_{n} = \frac{v}{λ _{n}} = n \frac{v}{2 L}, n = 1, 2, 3, \dots

$f_{1}$ is the fundamental frequency, and all the others are whole multiples of it. The same scheme, with different conditions at the ends, describes organ pipes: those open at both ends have the same series of harmonics as a string, those closed at one end have only the odd harmonics — half the available frequencies, for the same length.

1. The first three harmonics of a string

A taut string of length $L = 1.2 m$ has a speed of propagation of transverse waves $v = 48 m/s$ . Find the wavelength and frequency of the first, second and third harmonics.

λ_{1} = 2 L = 2.4 m, f_{1} = \frac{v}{λ _{1}} = \frac{48}{2.4} = 20 Hz

λ_{2} = L = 1.2 m, f_{2} = 2 f_{1} = 40 Hz

λ_{3} = \frac{2 L}{3} \approx 0.80 m, f_{3} = 3 f_{1} = 60 Hz

You do not need to recalculate everything from scratch for every harmonic: once you have found $f_{1}$ , the others are simply $n \cdot f_{1}$ .

Beats

When two sound waves of very close frequencies $f_{1}$ and $f_{2}$ superpose, the perceived intensity pulses with a beat frequency

f_{beat} = ∣ f_{1} - f_{2} ∣

2. Two almost identical tuning forks

Two tuning forks emit $f_{1} = 440 Hz$ and $f_{2} = 444 Hz$ respectively. What beat frequency is heard, and how many beats are counted in $5.0 s$ ?

f_{beat} = ∣440 - 444∣ = 4 Hz, N = f_{beat} \cdot t = 4 \cdot 5.0 = 20 beats

Beats are the tool piano tuners use to tune an instrument without needing a measuring instrument: when the beats disappear, the two frequencies coincide.

Decibels: a logarithm earlier than expected

The sound intensity level is defined from the intensity $I$ (power per unit area) relative to a reference threshold $I_{0} = 1.0 \times 1 0^{- 12} W/ m^{2}$ , the threshold of hearing:

β = 10 lo g_{10} (\frac{I}{I _{0}})

Here you only need to know one thing about the base-10 logarithm:

lo g_{10} (1 0^{n}) = n

Nothing else is needed for this chapter — the full theory (properties, equations, inequalities) arrives in maths later, but this one property is already enough for the calculations the test asks for.

3. Sound level and distance

A loudspeaker emits a sound with intensity $I = 1.0 \times 1 0^{- 4} W/ m^{2}$ at the point where you are. Calculate the level in decibels, then find how it changes if you move away until you double the distance from the source (use $lo g_{10} 2 \approx 0.301$ ).

β = 10 lo g_{10} (\frac{1.0 \times 1 0 ^{- 4}}{1.0 \times 1 0 ^{- 12}}) = 10 lo g_{10} (1 0^{8}) = 80 dB

With spherical propagation the intensity scales as $1/ r^{2}$ : doubling the distance brings the intensity down to a quarter, $I^{'} = I /4$ . The new level:

β^{'} = 10 lo g_{10} (\frac{I /4}{I _{0}}) = β - 10 lo g_{10} (4) = 80 - 10 \cdot 2 lo g_{10} (2) \approx 80 - 6.0 \approx 74 dB

The point the test checks on purpose: doubling the distance does not halve the level in decibels, it lowers it by only about 6 dB. It is the logarithmic scale that works this way — an idea that holds up the whole chapter even before you have seen logarithms in maths.

The Doppler effect

When source and observer move relative to each other, the perceived frequency changes:

f^{'} = f \frac{v \pm v _{O}}{v \mp v _{S}}

where $v$ is the speed of sound in the medium, $v_{O}$ that of the observer and $v_{S}$ that of the source; the signs are chosen by looking at whether the motion brings the two parties closer (higher perceived frequency) or further apart (lower). There is no single formula to learn by heart for each case: it is better to reason each time about who is approaching whom, and choose the sign accordingly — it is the point where most time is lost in tests.

A classic application: a stationary ambulance emits a siren at $f = 700 Hz$ and approaches a stationary pedestrian at $v_{S} = 25 m/s$ , with sound travelling in air at $v = 340 m/s$ . While approaching, the pedestrian hears a higher frequency:

f_{approaching}^{'} = f \frac{v}{v - v _{S}} = 700 \cdot \frac{340}{340 - 25} = 700 \cdot \frac{340}{315} \approx 756 Hz

An instant after passing, the source moves away at the same speed, and the frequency drops below the real one:

f_{receding}^{'} = f \frac{v}{v + v _{S}} = 700 \cdot \frac{340}{365} \approx 652 Hz

The perceived jump at the moment of passing — from $756$ to $652 Hz$ , over $100 Hz$ of difference — is exactly what you hear in the street when an ambulance passes: it is not an impression, it is the Doppler formula with the sign changing because the source goes from “approaching” to “receding”.

The mistakes that cost marks

Confusing the speed of propagation of the wave with the speed of the particles of the medium — they are two different things: in a sound wave the air does not “travel” towards the listener, it oscillates in place.
Using the open-pipe formula for a pipe closed at one end too — the relation between length and harmonics is different, and all the even harmonics are missing.
Thinking that doubling the intensity doubles the decibels — the scale is logarithmic, not linear: doubling the intensity adds only about 3 dB.
Getting the sign wrong in the Doppler effect because you did not draw who is moving towards whom before writing the formula.
Confusing beat frequency with mean frequency: the beat is the difference $∣ f_{1} - f_{2} ∣$ , not the average of the two frequencies.

How we work on this chapter

In lessons I always start from the drawing of the standing wave — how many nodes, how many antinodes, what fraction of $λ$ fits in the given length — before touching a formula: it is the quickest way to stop confusing string and open pipe with closed pipe. For decibels we work on the logarithm property seen above until it becomes automatic, without waiting for the full theory to arrive in maths. The picture of how I set up the physics course is on the page about physics tutoring.

The chapter that follows, from January to March, is electrostatics and circuits: I wrote the guide in electrostatics and circuits: preparing the Year 4 test. For the general picture of physics mistakes in high school, beyond the single chapter, there is the 5 typical physics mistakes in high school.

If you are at the classical high school, in the same year (Year 4) and in the same period, the physics syllabus is something else entirely: with only two hours a week the classical high school is still on circular motion and work-energy, a chapter that at the science high school is done in Year 3. You will find the corresponding guide in Year 4 physics at the classical high school: energy and circular motion.

The quickest way to find out where you stand before the test is the simulation on waves and sound: fifteen questions in fifty minutes, with the correct answer free and without registration; the guided step-by-step solution unlocks with the first lesson.

After mechanical waves, light behaves in the same way: superposition, path difference, fringes. It is the subject of interference and diffraction: Young’s experiment.

If you want a hand before the end-of-October test, you can start with a free 30-minute introductory call: write to me via contact.

Year 4 physics: mechanical waves and sound