The Boomy Music Room: Echo, Diffraction and What to Buy
A school is turning an 8.0 m long, 5.0 m wide classroom into a music room. The walls are bare concrete, the floor is vinyl on concrete, and the only opening is a 0.9 m wide doorway leading to a corridor. When the head of department claps sharply from one end wall, students describe a rapid flutter of repeats, and the room sounds boomy when the drum kit plays. A supplier offers three options: heavy curtains on the long walls ($900); 25 mm acoustic foam panels covering 40% of the wall area ($2400); or a solid door on the doorway plus carpet ($700). Separately, the physics class uses a 1.5 m plastic tube, closed at one end by a movable plunger and open at the other, driven by a signal generator and small speaker. At resonance with the plunger at the far end, the generator reads 57 Hz. The room air is at 20 degrees C, where the speed of sound is 343 m/s.
1. Use the clap to estimate the time delay of the reflection from the far wall, and explain why students hear a flutter of repeats rather than one clean echo.
Read the worked answer
The sound must travel to the far wall and back before the reflection returns:
- Path = 8.0 m + 8.0 m = 16 m
- t = d / v = 16 / 343 = 0.047 s
A second reflection, off the near wall behind the clapper, returns sooner, and successive bounces around the room arrive at roughly regular intervals of a few hundredths of a second. That is what produces the 'flutter' the students describe.
Why a flutter rather than a distinct echo? The ear fuses a direct sound with reflections arriving within roughly 0.1 s into a single perceived event, using the first arrival to judge direction. At 0.047 s the reflection falls inside that window, so it is heard as a short rough tail rather than a separate repeat. The same effect is why the room sounds boomy: the reflected energy builds up and decays slowly because the bare concrete, vinyl and glass absorb very little sound. Adding absorption does not change the 0.047 s travel time; it reduces the level of each reflection so the tail dies away faster.
2. The class finds resonance at 57 Hz in the 1.5 m tube. Calculate the speed of sound from this result and comment on how well it agrees with the expected 343 m/s.
Read the worked answer
A tube closed at one end and open at the other resonates when the air column holds a quarter of a wavelength (the fundamental):
- L = lambda / 4, so lambda = 4L = 4 x 1.5 = 6.0 m
- v = f x lambda = 57 x 6.0 = 342 m/s
This agrees with 343 m/s to about 0.3%, which is a good result and consistent with air at roughly 20 degrees C. The speed of sound in air depends on temperature, rising by about 0.6 m/s per degree, so a room at 15 degrees C would give nearer 340 m/s and a cold morning would give a noticeably lower value.
Two sources of small error are worth naming. The open end is not a perfect pressure node: the air just outside also moves, so the effective length is slightly longer than 1.5 m, which would push the calculated speed up slightly. And finding resonance by ear is imprecise, because the peak is broad; a student may read 55 Hz or 59 Hz as the loudest point, changing the answer by several m/s. Repeating with the plunger at two or three positions and checking the pattern would give a more reliable value.
3. Recommend one of the three supplier options for the music room. Justify your choice using the physics of the complaints and the cost, and describe a simple test the school could use to check whether it worked.
Read the worked answer
The two complaints have different causes, so the treatment has to address both.
Boominess is a low-frequency problem. The drum kit's fundamental energy sits around 60 to 100 Hz, giving wavelengths of 3.4 to 5.7 m. Thin 25 mm foam absorbs mid and high frequencies well but is almost transparent to sound with a wavelength of several metres, so $2400 of foam covering 40% of the walls would barely touch the boom and would leave the room sounding harsh and dead at the top end. That is poor value.
Escape to the corridor is an aperture problem. The 0.9 m doorway is a direct path for sound, and low frequencies diffract through it strongly because their wavelength is much larger than the opening. A solid door closes that path at source.
Recommendation: the door plus carpet ($700). It is the cheapest option and it directly addresses the corridor leak, while the carpet adds mid and high absorption and cuts footfall and drum-kit splash. If boominess persists, spend the savings on thick curtains hung with an air gap on one long wall, or on deep corner absorbers, and move the kit away from the corner where low-frequency energy collects.
Test: measure before and after with a sound level meter at a fixed seat, playing the same short piece at the same volume setting, and repeat the clap test to see whether the flutter tail has shortened.