Q.Let the source propagate a sound wave whose intensity at a point (initially) is I. Suppose we consider a case when the amplitude of the sound wave is doubled and the frequency is reduced to one-fourth. Calculate the new intensity of sound at the same point.

From a Ripple to a Roar: What Makes Sound "Loud" and "High"?

Think of a calm lake. Drop a small pebble — you get a gentle ripple that fades quickly. Drop a big rock — you get tall waves that travel far and hit the shore with force. That difference in energy is the first intuition for intensity and loudness.

Now imagine two different ripples: one that rises and falls slowly (say, one wave every two seconds) and another that jiggles rapidly (ten waves per second). The slow one feels like a gentle swell; the fast one feels like a buzz. That difference in rate is the intuition for frequency and pitch.

Sound works exactly the same way. A sound wave is a travelling disturbance in air pressure. Two independent properties of that wave determine everything you hear:

1. How much energy the wave carries → determines loudness.
2. How fast the wave oscillates → determines pitch.

Intensity and Loudness

Intensity is a physical, measurable quantity. It is the power carried by the sound wave per unit area, measured in watts per square metre ($\text{W/m}^2$). A bigger amplitude of the pressure variation means more energy is being transferred, so intensity is higher.

Loudness is the subjective sensation your brain creates from that intensity. It is not the same thing. Your ear does not respond linearly — it compresses a huge range of intensities into a manageable range of perceived loudness. A sound that is ten times more intense does not sound ten times louder; it sounds only about twice as loud.

The key relationship: loudness increases with intensity, but logarithmically. This is why we use the decibel scale:

A whisper is about $10^{-10} \text{ W/m}^2$ (20 dB), normal conversation is $10^{-6} \text{ W/m}^2$ (60 dB), and a jet engine at close range is $10^2 \text{ W/m}^2$ (140 dB). That is a trillion-fold range in intensity, but only a 120 dB range in loudness.

Frequency and Pitch

Frequency is the physical property: the number of complete oscillations (cycles) the sound wave makes per second. Its unit is the hertz (Hz). A tuning fork marked 256 Hz makes 256 compressions and rarefactions of air every second.

Pitch is the subjective sensation — how "high" or "low" a note sounds to you. A flute produces a high pitch (high frequency, around 1 kHz), a bass drum produces a low pitch (low frequency, around 50 Hz).

The relationship is simpler than for loudness: pitch increases monotonically with frequency. Double the frequency, and the pitch goes up by one octave — a musical interval that sounds like a natural jump (e.g., from middle C to the next C above). …