# Why is sound produced when we hit a metal? [duplicate]

We all know that metal are sonorous and produce sound on hitting but why is sound produced

• I thought that it might be a way of energy released on inelastic collision but then why not heat as heat is also a form of energy that can be released
• Why not both? Are you certain both aren't produced? Jul 20, 2018 at 2:56
• No I am not certain if both are produced or not Jul 20, 2018 at 17:27

The inelastic collision (loss of kinetic energy) causes the metal to vibrate. This vibration is transferred into the air, creating a pressure (sound) wave our ears register as sound. Some of the vibration goes into warming up the metal and the air, as the vibration increases the temperature of each.

For a more detailed explanation, every material has a unique speed of sound which depends on its Young's modulus $Y$ (for solids) or equivalently on its bulk modulus $\beta$ for fluids. First consider the metal itself. It's Young's modulus is given by

$$Y = \frac{\rm stress}{\rm strain} = \frac{F/A}{\Delta L/L}.$$

By counting the molecules and making connections between the macro- and micro- level behaviors, you can derive the equivalent expression* for the modulus

$$Y = \frac{k_{\rm s,i}}{d}.$$

Here we model each atom as a "ball" of diameter $d$, each ball connected to each other with a spring of interatomic stiffness $k_{\rm s,i}$. When you hit the metal, you give some atoms a particular stretch $x$, and thus from Hooke's law with damping factor $2b$

$$m_{\rm atom} \ddot x = -k_{\rm s,i}x - 2b\dot x,$$ which gives the decaying solution

$$x(t) = Ae^{-bt}\sin\left(\omega t + \phi\right),$$ where $\phi$ is some phase angle and $\omega^2 = {\frac{k}{m}} + b^2$ is the atom's angular frequency.

Then, the speed of sound in the metal is given by

$$v_s = \sqrt{\frac{Y}{\rho_m}},$$

where $\rho_m$ is the density of the metal. Sound is a longitudinal wave, meaning its oscillation is in the same direction as its motion. See the below gif.

Connecting these, you can show that the speed of sound within the solid is

$$v_s = \sqrt{\frac{k_{\rm s,i}}{m_{\rm atom}}}d.$$

Now, let's say you impart some total energy $E_{\rm hit}$ onto the metal in the inelastic collision. All of your energy goes into its spring potential energy, plus some heat, i.e.

$$\frac{E_{\rm hit}}{n} = \frac{1}{2}k_{\rm s,i}x^2 + Q = \frac{1}{2}m_{\rm atom}v_s^2\left(\frac{x^2}{d^2}\right) + Q,$$

where I have wrapped the dampened/lost energy into the heat per molecule $Q$. It's hard to quantitatively explain the interplay between heat and sound because it depends on the situation. However, my expectation is that in general

$$\frac{1}{2}m_{\rm atom}v_s^2\left(\frac{x^2}{d^2}\right) >> Q.$$

After all, when you hit something, the sound is generally loud, but the temperature change isn't much for a single hit.

You can extend this result for the air using the bulk modulus $\beta$, but it is the same concept. The initial hit drives the longitudinal pressure wave.

*I have omitted this derivation from my answer, but I can if you need it.

• Can you please explain with some data like how much energy is converted into sound waves and how much goes to form heat if possible Jul 20, 2018 at 15:18
• A diagram showing the vibration process will also be appreciated Jul 20, 2018 at 15:19
• @RinkiDwivedi see my edited response.
– zh1
Jul 20, 2018 at 15:56

One of the least springy metals is probably lead which is a sound damper, and can be used to absorb inverse speaker cone sound energy inside a home made loudspesker cab.

It's an elementary question for this forum.

All resonating materials generate percussion sound: wood, drumskins, minerals(eg glass).

Metal plates have consonant harmonic sounds and musical periodic waves. Sound energy is generated when air is displaced, by harmonic vibration modes(1) in the metal.

So if the metal length propagates a wave at a preferential frequency of one period every 10ms, 5ms and 1ms, the sounds will be 100hz, 200hz, and 1000hz. The lowest harmonic being the "fundamental frequency".

The wave speed is a function of metal stiffness and spring caracteristics. Spring steel is used for tuning forks rather than aluminium and lead.

Gongs and symbals have irregular dents hammered into them to break the harmonic vibration geometry of a simple metal plate, and gongs vibrate in many modes simultaneously: the fundamental frequency lasts very briefly, the gong vibrations divide into dozens of atonal/dissonnant pseudo-sine waves that occupy a arbitrary and random frequency distribution.

Drums are dissonant waves, metals are consonant waves.

"Physical modelling" acousticians are musicians that try to emulate object physics to recreate instruments through reductionist /ObjOriented models of object physics.