# Echo Analysis in Iberian Bullfighting Arenas Through Objective Parameters and Acoustic Simulation 1
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To perceive such an echo, the total distance travelled by the reflected sound must be at least 17 m, which is derived by multiplying the speed of sound by the delay of 50 ms. It is very unlikely for an echo to occur in a closed room, with a greater probability in positions far from the sound source (lower direct sound) and close to a reflective surface, as verified by [PERSON] et al. [4]. These authors, using a 2D geometric tool (GeoGebra), analyse the reflections of two pairs of parallelepiped walls, where they confirm that the most probable echo risk zone is always the area farthest from the source and closest to the rear wall. As the size of the enclosure increases, and the greatest risk of echo formation occurs in places far from the emitter and close to the reflective rear wall. The same conclusions are obtained when the floor and ceiling are included and analysed with Catt Acoustic. This risk of echo increases when focalisation occurs, such as with concave curved surfaces with cylindrical or elliptical geometry [5; 6], as found in bullrings and Roman amphitheatres [7]. Likewise, in an architectural space without a roof, the density of reflections is lower, and [PERSON] [8] points out that the existence of echoes is therefore more possible. Based on the findings of [PERSON], a criterion was proposed by [PERSON] and [PERSON] [9], represented by constant percentage disturbance contours of the amplitude of reflections as a function of delay time relative to the direct sound. Furthermore, [PERSON] and [PERSON] [10] have proposed an echo detection criterion based on the time of the centre of gravity of the impulse response. The criteria have been investigated for both speech and music. It is assumed that the two opposing physical magnitudes of the delayed reflection, the sound pressure amplitude \(p\) and the delay time \(t\), are linked by being a constant product with the acoustic pressure with an exponent \(n\) or weighting factor to be determined. One objective criterion for room acoustics that adequately links both physical magnitudes is the centre of gravity time. The centre of gravity time of an impulse response (\(p(t)\)) normalised to its total energy is  
\[t_{\text{S}}(\tau)=\frac{\int_{0}^{\tau}tp^{n}(t)dt}{\int_{0}^{\tau}p^{n}(t)dt} \tag{1}\]  
where \(n=2\) for the traditional centre time formula, and \(\tau=\infty\) in the upper limit of the integral, [PERSON] [11]. For this criterion, a suitable value of n was found to be \(n=2/3\) for speech and \(n=1\) for music. Lastly, the echo criterion (Echo Criterion EK) is expressed by the relationship between the time of the strong reflection \(\tau\) and the temporal duration of the reflection \(\Delta\tau\) as  
\[\Delta t_{\text{S}}=t_{\text{S}}(\tau)-t_{\text{S}}(\tau-\Delta\tau) \tag{2}\]  
\[EK=max\frac{\Delta t_{\text{S}}}{\Delta\tau} \tag{3}\]  
For speech, \(n=2/3\) and \(\Delta\tau=9\) ms. For music, \(n=1\) and \(\Delta\tau=14\) ms.  
A disturbing echo for speech appears if EK exceeds a critical value of 1 at a time delay of s \(>50\) ms, then there is a 50% chance that a listener will detect a disturbing echo for the spoken word. If \(EK\) exceeds a critical value of 1.8 at a time delay of \(s>80\) ms, then there is a 50% chance that a listener will detect a disturbing echo for light music, while with \"heavy\" or \"dense\" music, this value is definitely higher. The maximum value of \(EK\)(s) after 50 or 80 ms is the Dietsch echo parameter for speech and music, respectively. Table 1 summarises all the information presented. Through simulated and auralised impulsive responses to music and speech, listening tests are audited in order to indicate the existence of strong echoes and reflections in concert halls and auditoriums.