You may be looking at the title and be thinking “this is clearly clickbait”. Well, you are wrong. You see, there is a place in the United Kingdom - England more specifically - where this natural phenomenon occurs all the time. In this post, I’m going to explain the physics behind this strange event as well as letting you know where you can find it.
The Cathedral
The ‘special place’ that I have been referencing is in fact St Paul’s Cathedral in London. This magnificent cathedral has stood for over 300 years and was built upon the original church that was dedicated to St Paul - an apostle who taught the gospel of Christ to the first-century world - over 1400 years ago. In the late 17th century, Sir Christopher Wren consecrated the cathedral and, in doing so, accidentally created a tourist attraction ever since. Have a look at St Paul's Cathedral below.
St Paul’s Cathedral is absolutely beautiful but we aren’t interested in the history of the cathedral in this post nor most of the actual cathedral for that matter. Rather, we are interested in what is aptly known as “The Whispering Gallery”. The cathedral is 111.3 metres in height, making it one of the tallest in the world of its kind. Fortunately, we ‘only’ need to climb 30 metres in order to reach “The Whispering Gallery” - however, this still comes out to be 257 steps.
Small Sidenote
This is a bit of a side note but it is still some very interesting physics; we can calculate the increase in gravitational potential energy that one would gain by climbing up to the gallery. According to the ONS (Office for National Statistics), the average mass of a man in the UK is about 83.6 kg. Thus, if an average UK male were to climb the 257 steps, they would have to do approximately 24600 Joules of work. I arrived at this figure simply by using the m-g-h formula for gravitational potential energy:
We aren’t done yet with this sidenote. The average muscle efficiency of the human body is between 18 and 26%. Let’s take the average of this interval - 22%. That means in order to do 24600 Joules of mechanical work when climbing up the stairs, an average person would use up around 112000 Joules of energy (this is just 24600 / 0.22)!
When you look at any food packet and see the calorie count, you are actually viewing the quantity of kilocalories in that said food (1 kilocalorie = 1000 calories). One Joule of energy is about 0.239 calories or, in our everyday language, 0.000239 kilocalories (kcal). Thus, 112000 joules of energy is approximately 27 kilocalories. So, in climbing up the flight of stairs, you’d burn off 27 “calories”. This is pretty disappointing considering a can of coke has 139 “calories”. Now that we have gotten through our physics sidenote, we can get back to the post!
The Mysterious “Whispering Gallery”
The “Whispering Gallery” is essentially a circular walkway and sits at the base of the dome structure. Have a look at the image below to get a better idea.
As long there is not too much background noise and you whisper along the curving wall, the whisper will travel along the wall and it can even be heard by someone more than 33 metres away; this is about 1 and a ½ tennis courts in length to put it in perspective. There are other whispering galleries all around the world. For example, the Barossa Reservoir in South Australia can carry sound up to 140 metres! But, how do they actually work: what is the actual science behind it?
The Science Behind “Whispering Galleries”
Lord Rayleigh, one of the most influential physicists of the 19th century, pondered about the phenomenon described in St Paul’s Cathedral and devised an explanation for it. Normally, when a wave is emitted from a point source, it travels in all directions, essentially forming a sphere. As the wave travels, the effective sphere spreads out, reducing the intensity of the wave at any point. The intensity of radiant energy transferred by a wave is defined as the power transferred per unit area. We can mathematically derive a proportionality relationship between the intensity of the wave and the distance travelled by the wave from its point source.
Using our standard definition for the intensity of a wave, we can easily derive what is formally known as the ‘Inverse-Square Law’. We can denote the power of the wave as P - we do not need to derive an expression for the power of the wave as it does not depend on the distance travelled, r, as it is the initial intensity of the wave. We can also create an expression for the area, A, of the ‘effective sphere’ created by the wave. We know that the surface area of a sphere is 4πR^2. Thus, we can create a mathematical expression for the intensity of the wave as seen below.
Thus, the intensity of a wave is inversely proportional to the square of the distance travelled, r, by the wave. Ergo, if, say, a sound wave travels 4 metres, its intensity will be 16 times smaller than its initial intensity; you can therefore see how quickly the intensity of a wave will reduce as it travels away from its point source. A whisper, due to its relatively low amplitude, will quickly become negligible as it travels due to this inverse-square law.
However, when it comes to the “Whispering Gallery” in St. Paul’s Cathedral, this inverse-square law does not apply, as explained by Lord Rayleigh. He explained the phenomenon of the travelling whisper sound waves with a series of specularly reflected sound waves. Specular reflection follows what is commonly known as the ‘law of reflection’. As can be seen below, the angle of incidence of the wave to the normal is equal to the angle of reflection of the wave. This occurs when the reflecting surface is smooth like the curved wall of the “Whispering Gallery”. On the other hand, when the surface is rough, the wave undergoes diffuse reflection, causing the angle of reflection to be different to the angle of incidence, as shown in the image below.
Due to the sound waves undergoing specular reflection and the circular shape of the gallery, the waves form chords of the circular gallery. This allows the waves to travel along the walls of the gallery, essentially clinging to the walls. The physics behind this phenomenon lies in the concept of resonance.
Resonance is a key concept that is commonly taught in High School. Thus, I am going to cover it quite extensively as it will help you to understand how a whisper gallery works and you will most likely cover it at school regardless. Resonance typically comes under the topic of simple harmonic motion (SHM). I am not going to cover SHM in great detail as otherwise we would be here all day. However, SHM occurs whenever the acceleration of an object is always proportional, and in the opposite direction, to the displacement of that said object. The most common example of this is a pendulum as shown below.
Assuming there are no external forces, this oscillating systems has a constant time period and will oscillate with a constant frequency. The frequency that such a system oscillates at, when no external forces are present, is known as the natural frequency of the system. A periodic force can be applied to the system at regular intervals - imagine pushing someone on a swing at regular intervals; it's the same thing. When a periodic force is applied to an oscillating system, the response depends on the frequency of the periodic force. The system therefore undergoes forced oscillations, when a periodic force is applied to it.
The total effect on the oscillating system, as a result of this periodic force, depends on the frequency of this applied periodic force. The frequency of the system under the influence of a periodic force is known as the applied frequency. The response of a system, as a result of a periodic force being applied to it, is usually measured from the amplitude of oscillations of the system. Thus, physicists usually plot a graph of the variation of the system's amplitude against frequency, as shown below.
As you increase the applied frequency from zero, the amplitude of oscillations of the system increases until it reaches a maximum amplitude at a particular frequency (as marked in the image above), and then the amplitude decreases again. The phase difference between the displacement and the periodic force increases from zero to π/2 at the maximum amplitude, then from π/2 to π as the frequency increases further.
When the system is oscillating at the maximum amplitude, the phase difference between the displacement and the periodic force is π/2. The periodic force is then exactly in phase with the velocity of the system, and the system is in resonance. The frequency at which the maximum amplitude occurs is known as the resonant frequency. For an oscillating system with little or no damping (just think of damping as the degree of external forces applied to the system), at resonance, the applied frequency of the periodic force = the natural frequency of the system.
When objects oscillate in resonance, they can form very noticeable effects. A good example of this - apart from our "Whispering Gallery" - is the Tacoma Narrows Bridge. This bridge unfortunately collapsed completely in 1940 within the United States of America. The reason why this occurred, however, fundamentally relies on the concept of resonance. A crosswind caused a periodic force on the bridge span. The wind speed, incidentally, was such that the applied frequency was equal to the natural frequency of the bridge span. This caused resonance to occur, resulting in massive amplitudes of oscillations by the bridge, eventually causing the bridge to collapse as shown in the image below.
You may be wondering how this can apply to the "Whispering Gallery". You see, every object vibrates at a certain frequency, depending on its temperature and many other properties about that object (think size, structure, et cetera). Thus, if you apply a certain periodic force to that object, you can make the object oscillate at resonance with whatever force you are applying to that said object.
Moving back to our circular gallery, the concept of resonance is at the heart of why a whisper can travel so far. The average frequency of a human whisper is close to the natural frequency of the wall, resulting in the wall vibrating at resonance with your whisper. This is also why this phenomenon can only occur at certain frequencies. However, as aforementioned, the circular shape of the wall is key to this phenomenon as it allows these forced vibrations to essentially travel all around the wall, forming regular intervals of high and low pressure zones.
The red and blue zones represent higher and lower air pressures, respectively. Obviously, the difference is very, very small and unnoticeable to a human being. However, this causes a pattern to form - known as several modes forming - as shown in the diagram above. What is so special about this pattern is that, as can be mathematically proven, the intensity of the sound wave will no longer decay as the inverse of the distance travelled squared (1/r^2) but rather will simply only decay as the inverse of the distance (1/r).
Consequently, by the time the whisper reaches the other side of the circular gallery, it's intensity is much greater than if the sound wave obeyed the normal inverse-square law - such as, when you normally whisper in a room. The actual mathematical derivation of this phenomenon is far beyond the specifications of any High School Physics and will thus not be included. But, that being said, if you are interested in reading Lord Rayleigh's paper on this, you are more than welcome doing so. I have included a link to the original paper below - it is an interesting read!
Lord Rayleigh's Paper on 'The Problem of the Whispering Gallery': https://zenodo.org/record/1522134#.X2CHDmhKiHs
That is it for this post! I hope that you enjoyed it. Perhaps, you can try and visit your closest "Whispering Gallery" - they are found all around the world! If you have any questions, feel free to leave them in the comments down below. Also, if you have any recommendations for new posts, I am always up to creating blog posts round your suggestions. Take care!
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