Lesson 17 Resonance

When a force is applied repeatedly at the natural frequency of any system, large amplitude oscillations result, this is the phenomenon of resonance.

On July 1^st, 1940, a delegation of citizens met in Washington states to engineer the Tacoma Narrows Bridge. This bridge was the longest suspension bridge in the world, but was destroyed due to the force of resonance. The word resonance itself is the Latin noun for “echo.” Below is footage of the Tacoma Narrows Bridge:

With music, the pleasant sounds are strictly due to the resonance of the instrument. All instruments resonate equally no matter what note is played. On the violin or cello, the strings vibrate hundreds to thousands of cycles per second and cause the whole instrument to vibrate at the same frequency. Every vibration gives the instrument’s “sound box” a push so that the “sound box” vibrates resonantly. The vibrations excite resonant vibrations of the air inside the sound box.

A tuning fork is a common tool used to tune the piano. How does this device function? The tuning fork sets air molecules into harmonic motion. Air works as an elastic medium that is a good conductor of traveling oscillations waves. The tuning fork pushes air molecules into waves to the natural frequency of the tuning fork. When the pitch of the piano is equal to the pitch of the tuning fork, the waves from the piano can disturb its Naturally oscillating systems. Below is a video of a tuning fork being used to tune a piano:

Resonance obeys the differential equations of a mass on a spring:

F = -k x

m a = -k x

m (d^2x/dt^2) = -k/m x                              x = c sin (omega(0) t)

dx/dt = (omega(0)) c cos(omega(0) t)

d^2x/dt^2 = (-omega(0)^2) c sin (omega(0) t)

(-omega(0)^2) c sin (omega(0) t) = -k/m (c sin (omega(0) t))

omega(0)^2 = k/m

Due to the repetitive force applied while “pumping” on a swing, this motion is considered to be a form of resonance.

F = -k x                

F(0)sin (omega t) where the force applied at a different frequency is proportional to F

F =  -k x + F(0)sin (omega (t))

m a = -k x + m a(0) sin (omega(t))

d^2x/dt^2 = -kx/m + m/m a(0) sin (omega (t)) = -(omega(0)^2 x + a(0) sin (omega (t))

x = c sin (omega(0) t) + A sin (omega (t))

d^2x/dt^2 = -(omega(0)^2 c sin (omega(0) (t))) – (omega^2 A sin (omega (t)))

-(omega(0)^2 c sin (omega(0) (t))) – (omega^2 A sin (omega (t))) = -(omega(0)^2 c sin (omega(0) (t))) – (omega^2 A sin (omega (t))) + a(0) sin (omega (t))

A sin (omega (t)) = a(0)/ (omega(0)^2 – (omega)^2) sin (omega (t))

The size of the oscillations depends on the size of the force a(0) and how close its frequency (omega) is to the natural frequency omega(0).

The amplitude of forces oscillations is sensitive to the frequency.

A = a(0)/ (omega(0)^2 – (omega)^2)

If the frequency of an object is close to its natural frequency, spectacular things can happen. On of the most famous is the shattering of a wine glass due to the frequency of a persons voice. Here is a video that demonstrates the shattering of a glass:

It is important to note that glass is a viscous fluid. If glass was not viscous, there would be no such thing as a glass window. Forces can be sent out at many frequencies. One of the most detrimental forces are earthquakes. The force of earthquakes resonate at a very low frequency that causes buildings to shake and deteriorate as shown below:


Aoelian harps function at certain wind speeds, which cause vorteces to peel off at the relative frequency of the wire. The "sea organ" below functions similarily:


Most systems containing vorteces are all examples of resonance, an important aspect of physics.