the time reverberation of an empty hall without and with 500 audience is 1.5s and 1.4 s find the reverberation time with 800audience in the hall

To find the reverberation time of the hall with 800 audience members, we can use the concept of the Sabine formula, which relates the reverberation time (RT) to the volume of the hall and the absorption characteristics of the surfaces within it. The presence of people in the hall increases the absorption of sound, thus affecting the reverberation time.

Understanding the Problem

We know the following:

• Reverberation time without audience (RT₀): 1.5 seconds
• Reverberation time with 500 audience (RT₅₀₀): 1.4 seconds

We need to find the reverberation time with 800 audience members (RT₈₀₀).

Using the Absorption Concept

The reverberation time can be approximated using the formula:

RT = V / (A + A₀)

Where:

• V = Volume of the hall
• A = Total absorption of the hall surfaces
• A₀ = Absorption due to the audience

Calculating the Change in Absorption

When the audience increases, the absorption increases as well. We can denote the absorption due to the audience as A₀. The change in reverberation time can be related to the change in audience size. The difference in reverberation time when moving from no audience to 500 can help us understand how much absorption is contributed by the audience.

The change in reverberation time from no audience to 500 can be expressed as:

ΔRT = RT₀ - RT₅₀₀ = 1.5 - 1.4 = 0.1 seconds

Estimating Absorption for 800 Audience Members

Assuming that the absorption due to the audience increases linearly, we can find the absorption for 800 audience members. If we denote the absorption per person as a constant, we can set up a proportion:

Let the absorption for 500 audience members be A₅₀₀ and for 800 audience members be A₈₀₀.

We can express this as:

A₈₀₀ = (800/500) * A₅₀₀ = 1.6 * A₅₀₀

Finding the New Reverberation Time

Now, we can estimate the new reverberation time with 800 audience members. The change in absorption will affect the reverberation time:

RT₈₀₀ = V / (A + A₈₀₀)

Since we don't have the exact values for V and A, we can use the known reverberation times to estimate:

Since the change in absorption from 500 to 800 is proportional, we can assume:

RT₈₀₀ = RT₅₀₀ - (ΔRT * (800 - 500) / 500)

Substituting the values:

RT₈₀₀ = 1.4 - (0.1 * (300 / 500))

RT₈₀₀ = 1.4 - 0.06 = 1.34 seconds

Final Result

Thus, the estimated reverberation time with 800 audience members in the hall is approximately 1.34 seconds. This calculation assumes a linear relationship between audience size and absorption, which is a reasonable approximation for this scenario.