How to calculate a delay tower

Björgvin Benediktsson on May 11th 2011

When you are setting up for a large live sound show it’s inevitable to use delay towers. Delay towers help propagate the sound coming out of the P.A. making the concert reach the outer edges of the arena with maximum sound quality.
However, there is some math involved when it comes to calculating the delay in sound that a particular delay tower needs. Since sound travels at a specific speed we need to match the sound coming from the delay tower with the sound coming from the stage. Let’s look at one of the most practical applications mathematics can help you in your live sound job.

The Speed of Sound Equation

I’m using the metric system here, but feel free to convert the numbers over to imperial if you have a hard time with it. Sound travels at a particular speed. Just like with anything in physics it takes sound a while to get from point A to point B. Sound travels extremely fast so this isn’t a particular concern most of the time, but it can be good to know just how fast it travels and how you can take advantage of it to make your live gig sound better.
The equation for the speed of sound is as follows:

C = 331.45 + (0.597 x t)

C is the speed of sound.
331.45 is the speed of sound at sea level.
0.597 is the variation in the speed of sound depending on temperature.
t is the temperature in degrees. Basically how warm it is outside.
Now, usually we can give a ballpark figure of the speed of sound being 344 m/s, which is the speed of sound at 21°. Plugging our temperature into our linear equation we get an easy 344.

331.45 + (0.597 * 21) = 343.987 or approximately 344.

Now, since I live in the desert and a cool 21° is just plain wishful thinking I would have to account for a different speed of sound. An average here is probably around 31 so the sound here in Tucson travels faster than it would somewhere else.

331.45 + (0.597 * 31) = 349.957

This means that given a temperature of 31° Centigrade sound travels 5 m/s faster than it would in a milder climate.

Now, What Are the Practicalities of Knowing This?

I know, this is great and all but what’s the point. Why learn math at all if it doesn’t have a practical application. I hear you. A great application to use these calculations is when you are doing a live show in a big venue. If a venue is big enough the back will need additional delay towers to make everyone in the audience hear the music at the same time.
You see, if you are at the front you hear the music instantly because you are so close to the source and the P.A. system that’s at the stage. But if you are WAY in the back you need additional speaker systems to keep the music loud. That’s why there are delay towers at big concerts and live music festivals. Because sound intensity greatly diminishes over longer distances, with a 6 dB loss of “volume” every time we double the distance of the sound source we need delay towers to keep the sound intensity going.
But you need to be aware of the way sound travels if you want the sound coming from the delay towers to match the sound coming from the P.A. speakers at the front. Sound travelling from the stage is travelling at the speed of sound. But the sound travelling to our delay tower is travelling through electronics and is travelling much faster than our stage sound. Therefore we need to set the correct delay to our delay tower to match up the tower with our original signal. If we didn’t do this we’d get an unnatural repeat effect, where the signal from our delay tower blasts our the music before the stage sound reaches our ears.

Calculating Delay for a Delay Tower

Now, in order to calculate the correct delay for our tower we must use the speed of sound equation and take into account how far we are going to put our tower. Say we decided that out original sound from the stage was relatively weak at 30 meters. We want to set up a tower there to reinforce the stage sound but how much delay should we put on the tower?
We can use the following equation:

where v is the delay time
x is the distance between the sound source and the delay (30m in this case)
t is the speed of sound
By plugging in our variables we can easily find the correct delay time to align our tower with our P.A.
30/344 = 0.087 seconds, or 87 millisecond delay.

But we’re not quite done.

Now we have a perfectly aligned delay tower that’s blasting music into our ears where we are sitting on the grass 40 meters away or so. But it’s still a little unnatural to hear everything coming only from the delay tower. You don’ really feel like you’re watching the concert. So now we have to trick our brain into thinking that all the sound is coming from the stage. By adding just a little bit of extra delay to the delay tower we get the initial wave of sound from the stage before it’s reinforced by our delay tower.
Adding an extra 10 to 15 millisecond delay to our already delayed tower we get the feeling that our music is coming from the stage, and our delay tower is just helping to reinforce the signal from the P.A. Now your brain thinks that the sound is coming from the direction of the stage, and not the delay tower. We would end up with a delay tower of around 97 to 103 milliseconds.

Accounting for Inconsistencies

My friend went to an outdoor metal concert in Phoenix last summer. The temperature was so high the bands were flabbergasted as to why the audience was able to stand the immense Phoenix heat during the summer. When you’re dealing with concerts in the desert heat temperatures up to 40° (104°F) are not uncommon. So if you are the sound engineer responsible for the delay tower you must take your new variables into account.
With the temperature being 40°C the speed of sound has become much faster, or 331.45 + 0.597*40 = 355.33 or approximately 355 m/s. That’s more than 10 meters per second faster than before!
If we were putting up the same delay tower as before our delay would be different.
We would be dividing our distance of 30 meters by our newly acquired speed of sound, 355 m/s. 30/355 = 0.0845 seconds, or 84 milliseconds. Now, that’s not a whole lot of difference since a millisecond is so fast that it almost doesn’t matter. What matters is being aware of the different variables and factors that you need to take into account whenever you are working in live sound. You can use 344 m/s as a default speed of sound and usually get away with it, but isn’t it better to know how sound works instead of using some number without really knowing why?

Conclusion

I hope the minimal math involved in this tutorial didn’t deter you to learn the practical things you can do with it. The simple math above is easy enough to use, and very practical when it comes to making the most of your live sound setup.