Acoustic properties · Stiffness and inertia

Specific stiffness

How much stiffness is available for the mass carried?

Specific modulus E′/ρ compares an elastic property against density. Its square root gives the ideal slender-rod longitudinal wave speed in the selected direction.

Try the interactive demo

Start with something familiar

A strong motor in a very heavy vehicle does not automatically produce quick acceleration. For a vibrating material, elastic resistance and inertia also compete. Specific stiffness compares the two instead of looking at E alone.

E′ = 12 GPa

ρ = 400 kg/m³ · c ≈ 5477 m/s

E′ = 24 GPa

ρ = 800 kg/m³ · c ≈ 5477 m/s

Diagram · Two materials can have different modulus and density yet the same ratio. Equal ratios do not imply equal blank mass or equal radiation ratio.

Notice what changes
Modulus describes resistance to elastic strain; density describes mass per volume. At equal geometry, raising the first and raising the second push vibration frequency scales in opposite directions.
Understand why
Dividing E′ by ρ asks how much elastic resistance accompanies the mass burden. The ratio can stay the same even when both original quantities change.
Connect it to wood and making
For the ideal axial rod, the square root becomes wave speed. In a plate, geometry and the other material directions still intervene; the ratio is a useful comparison, not the whole instrument.

Put names to the ideas

Keep the dimensions

E′/ρ has units m²/s², not GPa and not a dimensionless score. Entering a modulus in GPa requires multiplying by 10⁹ before using SI density.

A ratio, not a new test

Once E′ and ρ are measured, specific modulus and the corresponding c follow algebraically. Displaying all of them does not create independent evidence.

Geometry still matters

Under fixed geometry, changing E′/ρ shifts modal frequency scales. Matching a finished plate’s frequency or mass is a separate problem involving thickness, arching and boundary conditions.

\frac{E'}{\rho}=c_L^2,\qquad [E'/\rho]=\mathrm{m^2/s^2}

E′/ρ is “elastic modulus per density.” If both double, their ratio and the derived axial speed stay unchanged. The final values have units, unlike a rating out of 100.

A worked example

At E′ = 12 GPa and ρ = 400 kg/m³, E′/ρ = 30 × 10⁶ m²/s² and c ≈ 5477 m/s. Doubling both E′ and ρ leaves c unchanged.

Explore it yourself

Try doubling modulus and density together, then change only one input. Watch which derived indicators remain unchanged.

Density kg/m³

Dynamic modulus GPa

Loss factor

c: 5345 m/s
E′/ρ: 28.57 10⁶ m²/s²
R: 12.73 m⁴/(kg·s)
Z: 2.24 MRayl
Q ≈ 1/η: 100.0
ACE: 1273 m⁴/(kg·s)

E′ + ρ → c → R, Z η → Q → ACE

Check your understanding

If E′ and ρ both double, what happens to specific stiffness?It doublesIt stays the same