Maths of the Sydney Opera House: Acoustics, Mathematics to Make Your Ears Ring


Copyright © Sydney Opera House Trust.

About this video

Are your ears ringing? Deconstruct the science of acoustics and sound with superstar maths teacher Eddie Woo and Mitch Allen, Soundscape Designer from the engineering firm Arup, whose founder worked directly with Utzon on the structure of the Opera House. 

Maths of the Sydney Opera House is a four part video series with award-winning maths teacher, author and Wootube star Eddie Woo uncovering the mathematical concepts behind the construction and design solutions of Opera House.

Watch other episodes in this series:



- Listen carefully to the sound of my voice. Come a bit closer. I want to share a secret with you. That secret is that everything you're hearing right now is governed by the laws of mathematics. And the mathematics of your sonic experiences can literally make your ears ring. But what's the relationship between maths and sound? If you were designing an art gallery to display majestic portraits and classic paintings, you'd need to think carefully about light. People wouldn't be able to see all that beauty and craftsmanship without a space that was appropriately illuminated. You'd need to consider the directions and intensity of natural light, as well as any artificial lights that would need to be added. The Sydney Opera House is also an art gallery of sorts, but it's primary focus is less on what you can see and more on what you can hear. And that's why today we're gonna talk about the mathematics of sound and acoustics. The musical instruments that perform here at the Sydney Opera House, come in all shapes and sizes. That's part of what gives each its signature sound. This acoustic guitar for instance, makes and projects unique sounds through the specific shape or geometry of its body. Let's see if an acoustic engineer can tell us how the specific shapes of theaters affect our experience of sound inside them. I wanted to ask you first a really general question about acoustics in theaters. What are the main factors that affect our experience of sound in spaces like those?

- There's three main things to pay attention to when designing a performance space. One is the background noise. So how much sound the mechanical services makes or the air conditioning. Another one is sound insulation. How much sound travels across a wall or a floor or the shells of the opera house. And room acoustics would be the third one. How the sound bounces around within that space and reflects off the surfaces of that space.

- Mitch, right now the concert hall is being renovated, and I noticed these panels are going up on the walls. What are these? And why are they such a funny shape?

- This is a really cool shape. And I love looking at it. The panels in the concert hall very much serve that purpose of reflecting the sound. So how the sound bounces around within that space. You might be able to see, it grades from a flat surface through to a wavy surface.

- Active architecture is designed to enhance audience experience. But enhancement means different things for different kinds of performance. If we were listening to a symphony orchestra, we would want the music to linger in the air, which means a larger space to allow the lush sound to reverberate for longer time. On the other hand, if we were trying to hear dialogue between two characters in a play, we would want greater clarity, which means a shorter reverberation. The amount of reverberation can be calculated precisely using a Sabine's formula, where V is the amount of volume in the space, S is the area of each surface in that space, and alpha is the number that tells us how much each of those surfaces absorb sound. We often think of an auditorium as a place to hear performers and their musical instruments. But the architects and engineers of the Sydney Opera House designed the auditorium itself as a musical instrument. And as with any instrument, performers need to know how to play it for the best effect. Otherwise, you can get some very unusual sounds. And I can show you a perfect example of one here in the Playhouse. This space is designed so that speech from the characters can come from the stage into the audience. But if we reverse that and make a sound from here among the seats, if you stand in just the right spot, listen to the result. This amazing echo comes from what we call specular reflections. Sound travels at 331 meters per second, but it has to travel different distances and therefore different amounts of time before bouncing back to our ears. This causes each returning sound to arrive with a slight delay, which we can hear as that amazing echo. Sound is front and center when you come to enjoy a performance here at the Sydney Opera House. But even with our full attention, it can be easy to miss some of the amazing mathematics and design that contribute to making every experience here a sonic delight.

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