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I have tried making an analogy with a simple pendulum: if you give it a push it will probably make several oscillations. But why isn't there just one ripple that would carry the energy of the several ripples?

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When the rock hits the water, a ripple is created and a volume of water is displaced. When the rock is submerged, the displaced water rushes back in to fill the void and collides at the centre, sometimes sending a column of displaced water into the air. This creates more ripples, especially when the vertical column, if there is one, falls back and creates still more ripples. Then turbulence follows for several seconds, sometimes accompanied by bubbles of air which have been sucked down into the void, creating yet more ripples, so it's not surprising that you have many ripples rather than one. The amount of ripples created is proportionate to the size, weight and shape of the rock and the speed at which it hits the water.

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The surface of the water acts like an elastic bedsheet, if you throw a stone at it the bedsheet will create vibrations and will carry on after the impact of the stone. This situation applies to a ponds surface as it acts like the bedsheet the only difference is that the stone sinks after the first impact with the water. The surface tension of the water is the reason that this happens and it happens in every situation when there is a contact between two objects but it can be so minute that it might not be visible to the human eye.

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A way to look at it is this: Every wave consists of a large number of wave components of various pure wavelengths. A very long continuous train of identical sinusoidal waves almost entirely consists of a single wavelength component.

An impulse like that of a rock hitting the water produces an initial disturbance that is highly localized. If we mathematically decompose that disturbance into a set of single-wavelength waves, we find that it is actually composed of a continuous spectrum of waves having a wide range of wavelengths, with each wavelength component itself comprising an infinite train of sinusoidal waves.

That may seem odd, but it works because at the moment the initial disturbance is created, all those waves are on top of each other and they cancel out except at the location of the disturbance where their crests are all precisely aligned. At that location, they add up to form the shape of the initial disturbance.

The speed of a surface wave in water depends on its wavelength: longer wavelength waves travel faster than shorter wavelength waves. So, the wave components from the initial disturbance sort themselves out as they travel away from the center: the longer wavelength components rapidly get ahead of the shorter wavelength components, and the farther they go the more sorted out they become and the more of each wavelength component's wave train becomes visible.

Two other factors are important. First, short wavelength surface waves are attenuated much faster than long wavelength surface waves, so short waves don't travel very far. Second, the size of the initial disturbance determines the spectrum and magnitude of the wave spectrum. Toss a tiny pebble into the water, and the dominant wavelength of the surface waves it produces is roughly the size of the pebble's diameter. Toss a meter-wide boulder in the water, and the dominant produced waves will be on the order of a meter wavelength. If a large slab of rock falls into the ocean, it will produce a tsunami whose dominant surface waves are the size of the slab.

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